Evaluation of coherence factor for high Q data

Transcription

1 Evaluation of coherence factor for high Q data M. Sutton February 19, 19 This is a calculation based on a formula in Pusey s review article[1]. The version I use is from the chapter in Mill s book[]. Here the wavelength of the x-rays is λ with spread λ, its wavevector is k = π/λ, c is the speed of light and Q = k f k i, the difference between the final and incident wavevectors. 1 β( κ) = V E i e i κ ( r r 1 ) V V Γ(, r r 1 Q, ( r r 1 ) ) d r ck 1 d r. (1) This gives the speckle shape, κ specifies the displacement between two positions of the x-ray detector (or between two detectors). The form of the mutual coherence function is (also in the chapter): (x x 1 ) (z z 1 ); Γ(x 1, z 1, x, z, t) = V E i e h e v e t /τ, () where h and v are the transverse coherence lengths in the horizontal and vertical directions. For an unfocused beam h = λ R sample /(πσ h ) and v = λ R sample /(πσ v ) where R sample is source to sample distance and σ h, σ v are the horizontal and vertical source sizes (RMS). The longitudinal coherence length is l = cτ = λ /(π λ) = λ/(k λ). This is the form used in Ref. [3, 4] (see Appendix I). The contrast of an intensity distribution I is typically defined as (I(max) I(min))/(I(max) + I(min)). The value β = β() and is related to the contrast squared of the speckle pattern and will be called the coherence factor. Sometimes this coherence factor is loosely referred to as the contrast. First we will calculate the coherence factor and then estimate the speckle sizes. 1

2 Figure 1: Coordinate system to define angles 1 Calculation of the coherence factor The coherence factor is given by the integral β = V V (x x 1 ) (y y 1 ) e h e v e t/τ. (3) Note that t = Q ( r r 1 )/ck. First we duplicate the calculations presented in Ref. [3] Using the coordinate system shown in Fig. 1, we can write the incident wavevector, the out-going wavevector and therefore the diffraction vector as: ki = k (, 1, ) kf = k (sin θ cos ϕ, cos θ, sin θ sin ϕ) Q = k sin θ (cos θ cos ϕ, sin θ, cos θ sin ϕ). (4) We also note that we have k = k i = k f and Q = Q = k sin θ. Thus, Q ( r r 1 ) = k sin θ(cos θ cos ϕ(x x 1 ) sin θ(y y 1 )+cos θ sin ϕ(z z 1 )). (5) This simplifies for Q in either horizontal or vertical diffraction to be: k sin θ cos θ x k sin θ y k sin θ cos θ z k sin θ y ϕ = (horizontal) ϕ = 9 (vertical).

3 Using the above form for τ (with ϕ = ), Q t/τ = ( r r 1 ) ck = k λ = λ λ Q k c λ λ λ sin θ((x x 1 ) cos θ (y y 1 ) sin θ) 1 Q 4k x λ Q y (6) λ k for ϕ = and using sin θ equals Q/(k ) and cos θ = 1 (Q/(k )). We now define A and B so t/τ = (A x + B y)/. Explicitly, A = k λ λ B = 4k λ λ cos θ sin θ = λ λ Q 1 Q 4k sin θ = λ Q = Q λ k l k l Q k For ϕ = 9 we simply replace x by z. Also note that B = A tan θ. Now we must specify the volume over which to integrate, i.e. the scattering volume. We chose a volume which respects the x, y and z directions of the coordinates systems and is of height M (along z) and width L (along x) and depth W (along y). With this symmetry, and specializing to ϕ =, the integral in Eqn. 3 factorizes into a two dimensional integral in z 1 and z and into a four dimensional in x 1, x, y 1 and y. We write this as β=β z β r. We first evaluate β z = 1 (z z 1 ) M dz 1 dz e v. (7) Using appendix II to evaluate this integral gives β z = 1 M = v M z dz(m z) e v [ M M M πerf( ) + e v 1 v v ] (8) Evaluating β r is similar, β r = 1 L (W L) L W W dx 1 dx dy 1 (x x 1 ) dy e h e A(x x 1 )+B(y y 1 ) 3

4 = = 1 L (W L) 1 (W L) ( e x h L L W dx 1 dx dx(l x) W (x x 1 ) ( dy(w y)e h e A(x x 1 )+By dy(w y) [ e Ax+By + e Ax By + e Ax+By + e Ax By ]) + e A(x x 1 ) By) ) = L W x [ (W L) dx(l x) dy(w y)e h e Ax+By + e Ax By ]. (9) This can be straight forwardly evaluated numerically. A few comments are worth making about the longitudinal coherence. 1. The longitudinal coherence is determined by the frequency resolution of the incident beam. In Goodman[5] an explicit formula is given for the longitudinal coherence in terms of the spectrum.. For a single bounce monochromator the typical spectrum will have 1/δQ tails and an exponential coherence time distribution (a Lorentzian spectrum) makes sense. 3. In Ref. [3] it was pointed out that for the first harmonic of an undulator spectrum (pink beam), a calculation using the formula in Goodman, gives a coherence length which agrees with that from the Lorentzian spectrum used above. 4. To contrast to the long tails of a Lorentzian spectrum, it is interesting to compare calculations to a Gaussian spectrum (exp( t ) which is τ more compact. The formula for β r then becomes L W x ] (W L) dx(l x) dy(w y)e h [e (Ax+By) 4 + e (Ax By) 4. (1) Note that τ is related to the half width of a Lorentzian spectrum but is related to the standard deviation for the Gaussian spectrum, so some factor should be used when comparing them. This spectrum was used in Abernathy et al[6]. 5. A two bounce monochromator (or multi-bounce) will have reduced tails in the frequency spectrum and this should give larger longitudinal coherence lengths than estimated by a Lorentzian spectrum. 4

5 Y W γ 11 1 L y=w+tan( γ )x t γ y=tan( )x X x rays Figure : Definition of angle for the scattering region. We will now generalize the calculation to have the sample at an angle γ as shown in Fig.. We will use a parallelepiped for the scattering volume. Such a geometry will directly apply to measuring a thin crystalline film whose planar dimensions are much larger than the beam dimensions. For reflection, one could imagine that the aborption length could be treated as the thickness for the sample. This changes the limits of integration for β r in Eqn. 9 by replacing the lower limits of y i (i=1,) from to x i tan γ and the upper limits from W to W + x i tan γ. Defining u i = y i x i tan γ and changing the variables of integration from y i to u i allows the integrals to be performed in a very similar way to Eqn. 9. In the end we obtain the result that a simple redefinition of A will handle this situation. Explicitly, β r = = 1 L (W L) 1 (W L) L L x1 tan γ+w dx 1 dx x 1 tan γ L W W dx 1 dx du 1 dy 1 x tan γ+w x tan γ (x x 1 ) dy e h e A(x x 1 )+B(y y 1 ) (x x 1 ) du e h e A(x x 1 )+B(u +x tan γ u 1 x 1 tan γ) 5

6 = = 1 L (W L) (W L) L L W W dx 1 dx du 1 dx(l x) W x dy(w y)e h (x x 1 ) du e h e (A+B tan γ)(x x 1 )+B(u u 1 ) [ e (A+B tan γ)x+by + e (A+B tan γ)x By ]. (11) So if A is replaced by A = A+B tan γ, this reduces to the previous formula. It is worth specializing to three standard diffraction geometries. For SAXS, conventionally γ = and this is the case first worked out (A = A). For reflection, one rotates the sample with Q and γ = 9 θ (θ is half of θ). Using the definitions of A and B and since tan(9 θ) = 1/ tan θ, A = A + B tan(9 θ) = A A tan θ/ tan θ =. A rotation is also done for diffraction in transmission but has an extra 9 offset, so γ = θ and A = A + B tan( θ) = A + B /A or A = A/ cos θ. Finally, one needs to relate W to the thickness of the film t. It is easy to see the W = t/ cos γ = t/ sin θ for reflection and W = t/ sin(θ 9) = t/ cos θ for transmission. Although varying γ with Q is good for comparing contrast factors and speckle sizes versus Q, most experiments are done with a fixed incident angle. In this case γ = 9 θ incident for reflection and γ = θ incident for transmission. Also W is the distance along the incident beam path, this can be set to the shortest of W = 1/µ, where µ is the absorption or to W = t/ cos γ for a sample of thickness t. For reflection, Eqn 9 can be analytically evaluated to give: β r = = (W L) [ L x W dx(l x)e h B W 1 + e B W ( B W ) ] h L dy(w y) [e By ] [ L L L ] πerf( ) + e h 1 (1). h h Remember that B W = W l Q k and so β r is the product of a function in W/ l and one in L/ h. Fig 3 compares the coherence factor for the three geometries (in the horizontal). For this and succeeding figures, the reference parameters are M = 1. µm, L = 5. µm, W = 1. µm, λ = 1.65 Å and λ/λ = The source to sample distance is 55 meters, and the source size is 345. by 45. µm (HxV) giving coherence lengths of h = 4. µm by v = 3. µm. These are typical parameters for Undulator A at the Advanced Photon Source. 6

7 .8 SAXS Reflection Transmission.6 β Q(Å 1 ) Figure 3: Comparison of three standard geometries. Figure 3 suggests that transmission geometry is better than reflection geometry. Reducing the thickness of the sample increases the contrast of the reflection geometry more than the others so that at a thickness of 5. µm all three are comparable and for smaller thicknesses reflection geometry dominates. Figure 4 compares horizontal diffraction to vertical diffraction. It plots for both 5x1 and 1x5 slits in each configuration for transmission geometry. I don t understand the large effect of difference between the black and blue curves yet. A similar set of curves exist for reflection geometry except by Eqn 1 simultaneously exchanging L and h and M and v gives identical coherence factors. Figure 5 shows the effect of varying the dimensions of the scattering volume. Each dimension is varied separately with the others left at the values given above. Note how the sample thickness in reflection matches the sample width in transmission. Equation 9 mixs one transverse and the longitudinal coherence effects. This probably explains the difference between the black and blue curves above. The coherence factor is further reduced if the detector area is larger than the speckle size. We leave a discussion of this effect until after the discussion 7

8 .8 1.,3. in plane (V) 5.,3. in plane (V) 1.,4. in plane (H) 5.,4. in plane (H).6 β Q(Å 1 ) Figure 4: Comparison of horizontal and vertial diffraction. The curves are labeled by their slit size, coherence length and orientation of their scattering plane, 8

9 .6.4 β. 5 1 L,M,W(µm) Figure 5: Effects of sample dimensions on coherence factor at Q = 1Å 1. The red curves are for reflection and the blue curves are for transmission. The solid lines are for varying L, the dashed for W and the dotted curve for L. 9

10 of speckle sizes in the next session. 1

11 Calculation of the speckle sizes We can estimate the speckle sizes by calculating the second moment of Eqn. 1 with respect to κ. This is twice the coefficient of the κ term in the expansion of Eqn 1. The second moment δ can be considered the variance of a Gaussian approximation for the shape of a speckle (e κ /(δ ) ) or the just the first term of an expansion in κ. 1 As for β, it makes sense to choose ϕ = to simplify the integral. From the form of the convolution integral, it can be seen that an odd function integrates to zero. Thus, we only need consider the three diagonal parts of the second moment, z z, x x and y y. Also, we can factor the resulting integrals into z-terms and r-terms with one term simply being one of the β i integrals above. We get: δ zz = = = 1 M β z M 1 β z M 4 v β z M dz 1 dz (z z 1 ) (z z 1 ) e v dzz z (M z) e v [ M M M πerf( ) + e v 1 v v ], (13) δ xx = = 1 L β r (W L) β r (W L) L L W W dx 1 dx dy 1 dx(l x) W x dy(w y)x e h (x x 1 ) dy (x x 1 ) e h e A (x x 1 )+B(y y 1 ) [ e A x+by + e A x By ] (14) and a similar term with x replaced by y for δ yy. Appendix III has yorick programs to calculate these three values. To calculate the speckle size in the plane of the detector select κ perpendicular to k f. For horizontal diffraction, δ zz is then the speckle size for the out of scattering plane direction. To get the speckle size in the scattering plane we use the parameterization of an ellipse given by r(ψ) = ab (b cos(ψ)) + (a sin(ψ)). 1 Use: where a is the value for κ = ( ) d ae κ dκ δ κ= = a δ 11

12 For us a = δ yy and b = δ xx and then the line perpendicular to k o cuts the center of the ellipse at ψ = θ + 9. This gives δ rr = sin (θ) δ yy 1 + cos (θ) δ xx and δ rr is the speckle width in the scattering plane (radial). For area detectors, θ would be the angle by which the plane of the detector is tilted around ẑ. Finally, one must consider the effect of the finite detector area. If its area is bigger than the speckle size, the measured coherence factor of the speckle pattern will be reduced and the speckle widths will be increased. A reasonable approximation is to assume the effect is the same as would be obtained by convolving a Gaussian representing the detector response with a Gaussian representing the speckle shape. Thus the speckle width is the sum in quadrature, of the speckle size estimated above to the size of a pixel in reciprocal angstroms. An estimate of the detector contribution is k U/(R det α) where U is the pixel size, R det is detector sample distance and the factor α converts the pixel size into a standard deviation (rms). In the calculations of appendix A of Ref [4] suitable for SAXS, that is for square pixels, α = 6 =.45. Explicitly the measured speckle sizes i are: ( ) i = δii + k U ( ) k U = δ ii 1 +. (15) αr det δ ii αr det The convolution of two Gaussians leads to the measured coherence factor being: 1 1 β = β ( ) ( ). (16) 1 + k U δ zzαr det 1 + k U δ rrαr det Figure 6 compares transmission and reflection geometries for the case of horizontal diffraction. A detector pixel size of µm at 1 m is used for convoluting with δ ii (α = 6). The curves for SAXS geometries are not plotted but they match quite well the transmission geometies as they did for β above. Figures 7 and 8 compares horizontal and vertical diffraction geometries. Figures 9 and 1 compares scattering volume dimensions for horizontal diffraction in both reflection and transmission geometries. (REDO to use 1m det distance). Finally, the effect of finite detector area on the contrast factor is presented in Fig 11. 1

13 .7 (1 4 Å 1 ) Q(Å 1 ) Figure 6: Comparison of speckle widths for reflection (red) and transmission geometry (blue). Horizontal diffraction is plotted, the dashed lines are for vertical widths and solid lines for horizontal widths. 13

14 .7 (1 4 Å 1 ) Q(Å 1 ) (1 4 Å 1 ).5 Figure 7: Comparison of speckle widths for horizontal and vertical geometries in reflection. Solid lines are for widths in the scattering plane and dashed lines are outof-plane widths. Black lines are for the 5 µm slit in plane horizontal (short coherence length direction). Blue lines are for a 1 µm in plane horizontal. Green lines are for the 5 µ slit in a vertical scattering plane and blue lines are for the 1 µm direction in the vertical scattering plane Q(Å 1 ) Figure 8: Comparison of speckle widths for horizontal and vertical geometries in transmission. Coloured lines have the same meaning as for Fig 7. 14

15 .6 1. (1 4 Å 1 ).5.4 (1 4 Å 1 ) L(µm).4 Figure 9: Comparison of speckle widths as scattering volume dimensions are changed, horizontal diffraction and reflection geometry. Solids lines for varying L, dashed for W and dotted for M. Red is for vertical direction and blue for horizontal. The horizontal green line is the width from the size of a pixel in the CCD. 4 L(µm) Figure 1: Comparison of speckle widths as scattering volume dimensions are changed, horizontal diffraction and transmission geometry. Line definitions are as for Fig 9. 15

16 1. β,frac Q(Å 1 ) Figure 11: Effect of detector size on the coherence factor. The coherence factors correspond to the widths in Fig 6. Reflection geometries are in red and transmission in blue. The dashed curves are the coherence factors, the dotted curves are the fraction by which the coherence factor is reduced and the solid lines are the resulting coherence factors. 16

17 Random points: for SAXS, with larger wavelength spreads (pink beam) the speckle are elongated in the radial directions (thus the nomenclature β r, for ϕ = and 9 radial is x and z directions). Should be able to handle asymmetric diffraction as well (giving elongated speckles). 17

18 Figure 1: Integration regions for doing the integral. Appendix I: Frequency spectrum The form of the temporal coherence is e iωt e t /τ. The frequency spectrum is then: S(ω) = e iωt e iωt e t /τ τ dt = 1 + τ (ω ω ) From this it can be seen that the width of the frequency spectrum is ω = 1/τ and it is the HWHM. Another form of the temporal coherence is e iωt e t τg, then: S(ω) = e iωt e iωt e t τg dt = πτg e τ g (ω ω ). Thus the sigma width is σ ω = 1/τ g. Converting both to FWHM gives δω = (.35) /τ FWHM = 5.5/τ FWHM Appendix II: Evaluation of correlation integrals We wish to evaluate integrals of the form: F (M) = dx dyf(y x), (17) 18

19 which often arise in correlation type calculations. From Fig. 1, we can divide the region of integration into two triangular regions, A and B. In region A, first integrate over x and in region B, over y. This gives F (M) = y x dy dxf(y x) + dx dyf(y x). (18) Change the variable of integration so u = y x. Then dx is du in the first and dy = du in the second. Changing limits of integration and relabelling an integration variable gives the following: F (M) = = = dx dx dy y y x duf(u) + duf( u) + dx dx x x duf( u) duf(u) dy {f(y) + f( y). (19) One integral can be done by intergration by parts. Defining g(x) = x dyf(y) and using the two parts u = g(y) and dv = dy to become v = y and du = g (y)dy = f(y)dy gives Thus, we find dy g(x) = g(m)m F (M) = = M dy f(y) Appendix III: Yorick programs dy y g (y) dy y f(y)). () dy (M y) {f(y) + f( y). (1) Here is a yorick file which calculates β and δ. To numerically evaluate Eqn 9, 11 and 14 we simply divide the region of integration into a by grid. This gives sufficient accuracy for typical parameters used at the APS. These and other yorick functions can be found at the URL mark/coherence/yorick/. // Using Pusey s formula modified f o r phi=9 and gam non 9. // // to avoid c o n f l i c t with beta ( ) d e f i n e d in gamma. i 19

20 func bbeta (Q, L,W, x i i p,m, xi op, k, d l o l, gam, gauss =,twobounce=) / DOCUMENT bbeta (Q, L,W, x i i p,m, xi op, k, d l o l, gam, gauss =,, twobounce=) C a l c u l a t e beta f o r given Q s. L,M are s l i t s s i z e s ( in plane (H) x out of plane (V) ), W i s along beam d i r e c t i o n, W=t h i c k n e s s / cos (gam) f o r thin f i l m = 1/(mu) f o r a b s o r p t i on dominated. x i i p, x i o p are coherence l e n g t h s ( in plane x out of plane ). k i s pi /lambda. d l o l i s d e l t a lambda over lambda (HWHM). gam i s angle o f s u r f a c e and normal to i n c i d e n t beam ( r a d i a n s ) gam= f o r SAXS gam=pi/ t h e t a i n c i d e n t f o r r e f l e c t i o n ( or pi/ tth / ). gam= t h e t a i n c i d e n t f o r t r a n s m i s s i o n ( or tth / ). gam= pi/ tth / gauss=1 or twobounce=1 w i l l chose a d i f f e r e n t frequency spectrum from the d e f a u l t which i s Lorenztion ( r e s p e c t i v e l y Gaussian and a double bounce mono ). In plane and out of plane r e f e r to the s c a t t e r i n g plane. Note : t h i s was designed f o r h o r i z o n t a l d i f f r a c t i o n at a synchrotron as t y p i c a l l y i s done f o r COSAXS at 8ID I. / //NOTE: changed order o f eta and x i Dec/4 { extern tth, AA, BB; i=dimsof (Q) ( ) ; tth= asin (Q// k ) ; i f (numberof(gam)==1) gam=array (gam, i ) ; i f (numberof(w)==1) W=array (W, i ) ; z=array (., i ) ; BB=AA=z ; NUM=; // use a x g r i d f o r i n t e g r a l, should be good to 1% for ( j =1; j<=i ; j++) { // loop over Q s z ( j )= betar ( tth ( j ), gam( j ),L,W( j ), x i i p, k, d l o l, gauss=gauss, twobounce=twobounce ) ; AA( j )=A; BB( j )=B; z = betaz (M, x i o p ) ; return ( z ) ; func betaz (M, x i ) { / DOCUMENT betaz (M, x i )

21 c a l c u l a t e s the beta f o r the d i r e c t i o n p e r p e n d i c u l a r to the s c a t t e r i n g plane. x i=coherence sigma=lambda Rsource /( pi sigma ( source ) ). M i s the s i z e. Gaussian coherence. / require, gamma. i ; x=m/ x i ; t = ( x sqrt ( pi ) (1. e r f c ( x))+exp( x ˆ) 1.)/ x ˆ ; return ( t ) ; require, gamma. i ; func betar ( tth, gam, L,W, xi, k, d l o l, gauss =,twobounce=) { / DOCUMENT betar ( tth, gam, L,W, xi,q, k, d l o l ) c a l c u l a t e betar in s c a t t e r i n g plane by brute f o r c e. L i s t r a n s v e r s e dimension, W i s sample t h i c k n e s s. x i i s coherence length ( in microns ), Q i s wavevector and k i s pi /lambda ( angstroms ˆ 1) and d l o l i s d e l t a lambda over lambda (HWHM). keywords : gauss =, twobounce= / extern NUM; // l o c a l A,B, x, y extern A, B, x, y // need to automate mapping from d l o l (FWHM) to c o r r e c t f a c t o r //W=min ( 1.,W/ cos (gam ) ) ; //W=W/ cos (gam ) ; //Now handled in c a l l i n g r o u t i n e Q= k sin ( tth /. ) ; //A = d l o l Q 1 e4 s q r t (1. (Q/( k ) ) ˆ ) ; // convert Q to micronsˆ 1 //B = d l o l 1 e4 Qˆ/ k ; A = d l o l Q 1 e4 sqrt (1. (Q/( k ) ) ˆ ) ; // convert Q to micronsˆ 1 B = d l o l 1 e4 Qˆ/ k ; A = A + B tan (gam ) ; x=span (., L,NUM)( :1:NUM, ) ; y=span (.,W,NUM) (, : 1 :NUM) ; t =. (L x ) (W y ) / ( ( L W)ˆ) exp( ((x/ x i ) ˆ ) ) ; i f (! is void ( gauss ) ) { // c o r r e c t f o r h a l f widths A = // Gaussian B = // s q r t ( l o g ( ) ), convert HWHM t = exp(.5 (A x+b y)ˆ)+exp(.5 (A x B y ) ˆ ) ; else i f (! is void ( twobounce ) ) {//CHECK t h i s //A = // Lorentzian squared //B = // s q r t ( s q r t (.) 1.) t = (1.+ abs (A x+b y )/)ˆ exp( abs (A x+b y))+ 1

22 ( 1. + abs (A x B y )/)ˆ exp( abs (A x B y ) ) ; else { t = exp( abs (A x+b y ) ) + exp( abs (A x B y ) ) ; // Lorentzian return (sum( x ( 1, ) y (, 1 ) t ) ) ; func betarg (Q, L,W, xi, k, d l o l ) { / DOCUMENT betarg (Q, L,W, xi, k, d l o l ) c a l c u l a t e s the betar f o r r e f l e c t i o n geometry using a n a l y t i c form. L i s t r a n s v e r s e dimension, W i s sample t h i c k n e s s. x i i s coherence length ( in microns ), Q i s wavevector and k i s pi /lambda ( angstroms ˆ 1) and d l o l i s d e l t a lambda over lambda (HWHM). / require, gamma. i ; x=l/ x i ; t = ( x sqrt ( pi ) (1. e r f c ( x))+exp( x ˆ) 1.)/ x ˆ ; B = abs ( d l o l 1 e4 Qˆ/ k ) ; // note l a c k o f minus s i g n tth= asin (Q// k ) ; gam = pi /. tth /. ; //W=min ( 1.,W/ cos (gam ) ) ; //W=W/ cos (gam ) ; //Now handled in c a l l i n g r o u t i n e t = ( B W 1+exp( B W) ) / / (B W)ˆ return ( t ) ; func betargauss (A, B, x i i p, L,W) { zz =.5 x i i p /L/ sqrt (1.+(A x i i p )ˆ+4. (W B) ˆ ) ; return ( zz ) ; And here are some yorick functions for calculating speckle widths. Note that the function also returns the coherence factor as the integrals needed for it are calculated along the way. // Using Pusey s formula modified f o r phi=9 and omega non 9. // func d e l t a v (Q, L,W, x i i p,m, xi op, k, d l o l, gam, gauss =,twobounce=) / DOCUMENT d e l t a v (Q, L,W, x i i p,m, xi op, k, d l o l, gauss =,omega=,twobounce=) C a l c u l a t e 1/ d e l t a ˆ ( and betas ) f o r given Q s. L,M are s l i t s s i z e s ( in plane (H) x out of plane (V) ), W i s along beam d i r e c t i o n, W=t h i c k n e s s / cos (gam) f o r thin f i l m

23 = 1/(mu) f o r a b s o r p t i on dominated. x i i p, x i o p are coherence l e n g t h s ( in plane x out of plane ). k i s pi /lambda. d l o l i s d e l t a lambda over lambda (HWHM). gam i s angle o f s u r f a c e and normal to i n c i d e n t beam ( r a d i a n s ) gam= f o r SAXS gam=pi/ t h e t a i n c i d e n t f o r r e f l e c t i o n ( or pi/ tth / ). gam= t h e t a i n c i d e n t f o r t r a n s m i s s i o n ( or tth / ). gam= pi/ tth / gauss=1 or twobounce=1 w i l l chose a d i f f e r e n t frequency spectrum from the d e f a u l t which i s Lorenztion ( r e s p e c t i v e l y Gaussian and a double bounce mono ). In plane and out of plane r e f e r to the s c a t t e r i n g plane. Note : t h i s was designed f o r h o r i z o n t a l d i f f r a c t i o n at a synchrotron as t y p i c a l l y i s done f o r COSAXS at 8ID I. { NOTE: r e t u r n s a 4 by number o f Q matrix, f i r s t t h r e e rows are 1/ d e l t a i i ˆ ( inv angrstroms ), f o u r t h row i s beta. / extern tth, AA, BB; i=dimsof (Q) ( ) ; tth= asin (Q// k ) ; i f (numberof(gam)==1) gam=array (gam, i ) ; i f (numberof(w)==1) W=array (W, i ) ; z=array (., i ) ; d e l t a s=array (., 4, i ) ; NUM=; // note using i s good to 1% t =betaz (M, x i o p ) d =d e l t a z (M, x i o p ) s c a l e =1. e4 ; // convert to angstroms from microns for ( j =1; j<=i ; j++) { t= betar ( tth ( j ), gam( j ),L,W( j ), x i i p, k, d l o l, gauss=gauss, twobounce=twobounce ) ; // x, y d i r e c t i o n t1=d e l t a r ( tth ( j ), gam( j ),L,W( j ), x i i p, k, d l o l, 1.,., gauss=gauss, twobounce=twobounc t=d e l t a r ( tth ( j ), gam( j ),L,W( j ), x i i p, k, d l o l,., 1., gauss=gauss, twobounce=twobounc d e l t a s ( 1, j )= s c a l e ˆ t1 / t ; d e l t a s (, j )= s c a l e ˆ t / t ; d e l t a s ( 3, j )= s c a l e ˆ d/ t ; d e l t a s ( 4, j )=t t ; return ( d e l t a s ) ; func d e l t a z (M, x i ) 3

24 { / DOCUMENT d e l t a z (M, x i ) c a l c u l a t e s the 1/ d e l t a ˆ ( microns ˆ) f o r the d i r e c t i o n p e r p e n d i c u l a r to the s c a t t e r i n g plane. x i=coherence sigma=lambda Rsource /( pi sigma ( source ) ). M i s the s l i t s i z e. Gaussian coherence. / x=m/ x i ; t = x sqrt ( pi )/. (1. e r f c ( x))+exp( x ˆ) 1.; t = ( x i /x ) ˆ ; return ( t ) ; //1 e8 convert to inv angstroms ˆ func d e l t a r ( tth, gam, L,W, xi, k, d l o l, dx, dy, gauss =,twobounce=) { / DOCUMENT d e l t a r ( tth, gam, L,W, xi,q, k, d l o l ) c a l c u l a t e 1/ d e l t a r ˆ by brute f o r c e ( microns ˆ ). L i s t r a n s v e r s e dimension, W i s sample t h i c k n e s s. x i i s coherence length ( in microns ), Q i s wavevector and k i s pi /lambda ( angstroms ˆ 1) and d l o l i s d e l t a lambda over lambda (HWHM). keywords : gauss =, twobounce= / extern NUM; l o c a l A, B, x, y ; //W=min ( 1.,W/ cos (gam ) ) ; //W=W/ cos (gam ) ; //Now done in c a l l i n g r o u t i n e // c o r r e c t f o r h a l f widths Q= k sin ( tth /. ) ; A = d l o l Q 1 e4 sqrt (1. (Q/( k ) ) ˆ ) ; // convert Q to micronsˆ 1 B = d l o l 1 e4 Qˆ/ k ; A = A + B tan (gam ) ; x=span (., L,NUM)( :1:NUM, ) ; y=span (.,W,NUM) (, : 1 :NUM) ; // Should t h e r e be a?? t=(l x ) (W y ) / ( ( L W)ˆ) exp( ((x/ x i ) ˆ ) ) ; t =(dx x+dy y ) ˆ ; i f (! is void ( gauss ) ) { A = // Gaussian B = // s q r t ( l o g ( ) ), convert HWHM t = exp(.5 (A x+b y)ˆ)+exp(.5 (A x B y ) ˆ ) ; else i f (! is void ( twobounce ) ) { //CHECK THIS A = // Lorentzian squared B = // s q r t ( s q r t (.) 1.) t = (1.+ abs (A x+b y ))ˆ exp(. abs (A x+b y))+ ( 1. + abs (A x B y ))ˆ exp(. abs (A x B y ) ) ; 4

25 else { t = exp( abs (A x+b y))+exp( abs (A x B y ) ) ; // Lorentzian return ( sum( x ( 1, ) y (, 1 ) t ) ) ; 5

26 References [1] P. N. Pusey, Statistical Properties of Scattered Radiation, in Photon Correlation Spectroscopy and Velocimetry, Editors, H. Z. Cumming and E. R. Pike, (New York: Plenum) (1974). [] M. Sutton, Coherent X-ray Diffraction, Chapter in Third-Generation Hard X-ray Synchrotron Radiation Sources: Source Properties, Optics, and Experimental Techniques, edited by. Dennis M. Mills, John Wiley and Sons, Inc, New York, (). [3] A. R. Sandy, L. B. Lurio, S. G. J. Mochrie, A. Malik, G. B. Stephenson, J. F. Pelletier and M. Sutton, Design and Characterization of an Undulator Beamline Optimized for Small-Angle Coherent X-ray Scattering at the Advanced Photon Source, J. Synch. Rad., 6, 1174 (1999). [4] D. Lumma, L.B. Lurio, S.G.J. Mochrie and M. Sutton, Area detector based photon correlation in the regime of short data batches: data reduction for dynamic x-ray scattering, Rev. Sci. Inst. 71, (). [5] Goodman, J. W., Statistical Optics, John Wiley & Sons, New York, USA (1985). [6] D.L. Abernathy, S. Brauer, G. Grübel, I. McNulty, S.G.J. Mochrie, N. Mulders, A.R. Sandy, G.B. Stephenson, M. Sutton, Small-angle coherent x-ray scattering using undulator radiation at the ESRF. J. Synchrotron Rad., 5, (1998). 6