BornAgain User Manual - Scientific Computing Group
Contents
1. W L F L sinc 4 x T H 2 W 2 dz arccos 1 sinc T arccos r exp iq Z LWH V gt 7 S LW 60 Examples Figure B 30 Normalized intensity F V computed with L 25 nm W 10 nm H 8 nm for four different angles w of rotation around the z axis References Agrees with the Ripplel form factor of FitGISAXS 14 61 Ha V and B 15 Ripple2 saw tooth Real space geometry W W 2 Perspective Top view Side view Figure B 31 An infinite ripple with an asymmetric saw tooth profile Syntax and parameters FormFactorRipple2 length width height asymmetry with the parameters e length L e width W e height H e asymmetry d They must fulfill di lt W 2 Form factor etc L F LW sine x i dz 1 2 sine p 1 S plas gad ETT 62 LWH V 2 7 S LW Examples Pa V Figure B 32 Normalized intensity F V7 computed with L 25 nm W 10 nm H 8 nm and d 5 nm for four different angles w of rotation around the z axis The low symmetry requires other angular ranges than used in most other figures References Agrees with the Ripple2 form factor of FitGISAXS 14 63 B 16 Tetrahedron Real space geometry 4 Aum Perspective Top view Side view Figure B 33 A truncated tetrahedron Syntax and parameters FormFactorTetrahedron length height alpha with the parameters e length of one e2. U r y r 4a xl 2 16 This is practically always adequate for material investigations with X rays or neutrons where the aim is to deduce y r from the scattered intensity r Since detectors are always placed at positions r that are not illuminated by the incident beam we are only interested in the scattered wave field e Am LOG euc 2 17 2 2 2 Far field approximation We can further simplify 2 17 under the conditions of Fraunhofer diffraction the distance from the sample to the detector location r must be much larger than the size of the sample Since the scattered wave w r only depends on r through the Green function G r r we shall derive a far field approximation for the latter We choose the origin within the sample so that the integral in 2 17 runs over r with r lt r This allows us to expand 7 k 7 ee ae 2 18 F 2 Verification under the condition r 0 is a straightforward exercise in vector analysis For the special case r 0 one encloses the origin in a small sphere and integrates by means of the Gauss Ostrogadsky divergence theorem This explains the appearance of the factor 47 13 where we have introduced the outgoing wavevector r T We apply this to 2 14 and obtain in leading order the far field Green function elkr oe GTP T Arr We r 2 20 where pelr eer 2 21 is a plane wave propagating towards the detector and y designates the complex con
3. H V a 1 ie 3fph 12 TR H gt fR H lt R HP v Figure B 40 Normalized intensity F V computed with R 3 3 nm H 9 8 nm and fp 1 8 for four different tilt angles 0 rotation around the y axis References Agrees with the IsGISAXS form factor Sphere 7 Eq 2 33 or TruncatedSpheroid 8 Eq 228 13 Bibliography 1 MARIA Magnetic reflectometer with high incident angle http www mlz garching de maria 2 NREX Neutron reflectometer with X ray option http www mlz garching de nrex 3 REFSANS Horizontal TOF Reflectometer with GISANS option http www mlz garching de refsans 4 High Data Rate Processing and Analysis Initiative HDRI of the Helmholtz As sociation of German research centres http www pni hdri de 5 SINE2020 world class Science and Innovation with Neutrons in Europe in 2020 http cordis europa eu news rcn 124015_en html 6 R Lazzari J Appl Cryst 35 406 2002 7 R Lazzari IsGISAXS manual version 2 6 http www insp jussieu fr oxydes IsGISAXS figures doc manual html as per May 2015 8 G Renaud R Lazzari and F Leroy Surface Science Reports 64 255 2009 9 V P Sears Neutron Optics Oxford University Press Oxford 1989 10 M Lax Rev Mod Phys 23 287 1951 11 M Born and E Wolf Principles of Optics Cambridge University Press Cam bridge 1999 12 E Hecht Optics Addison Wesley San Francisco 2002
4. Solutions of this equation in spherical coordinates have a well known series expansion We send R so that we need only to retain the lowest order the form of which has been anticipated in the boundary condition 2 9 eiKR G r R v p Tg ep I p A 7 and similarly eiKR B r R v p rp Tap Ol p A 8 The functions g and b can be further expanded into spherical harmonics but this is of no interest here The decisive point is the factorization of G and B and their common R dependence It follows at once that I rs rp do R dependent bg gb A 9 OS From A 5 we obtain the reciprocity theorem G rp Ts Blrs rp A 10 It allows us to obtain the far field value of the forward propagating Green function G at the detector position rp from the adjoint Green function B that traces the radiation back from rp to the source location rg The theorem is practically important because B is much easier to compute than the unexpanded G 29 Appendix B Form factor library BornAgain comes with a comprehensive collection of hard coded shape transforms for standard particle geometries like spheres cylinders prisms pyramids or ripples This collection is documented in the following For each shape the real space geometry is shown in orthogonal projections the parameters of the BornAgain method are defined an analytical expression for the form factor is given and exemplary results for F q vers
5. form factor 44 ellipsoidal 46 Detector mapping the cross section 6 transmission geometry 25 Dissipation 26 Distorted wave Born approximation 5 18 19 multilayer 22 Download 8 18 DWBA see Distorted wave Born approximation Ellipsoid form factor truncated 50 Ellipsoidal cylinder form factor 46 Evanescent wave 26 Facetted cube form factor 66 Far field approximation 13 14 18 Fermi s pseudopotential 11 Flux incident and scattered 15 Form factor 73 FormFactorAnisoPyramid 34 FormFactorBox 36 FormFactorCone 38 FormFactorCone6 40 FormFactorCuboctahedron 42 FormFactorCylinder 44 FormFactorEllipsoidalCylinder 46 FormFactorFullSphere 48 FormFactorFullSpheroid 52 FormFactorHemiEllipsoid 50 FormFactorPrism3 54 FormFactorPrism6 56 FormFactorPyramid 58 FormFactorRipplei 60 FormFactorRipple2 62 FormFactorTetrahedron 64 FormFactorTruncatedCube 66 FormFactorTruncatedSphere 70 FormFactorTruncatedSpheroid 72 Forum 9 Fraunhofer approximation 13 Fresnel coefficients 18 23 Full sphere form factor 48 Full spheroid form factor 52 Glancing angle 16 Green function homogeneous material 13 14 reciprocity 28 vertically structured material 18 Helmholtz equation 12 Hemi ellipsoid form factor 50 Hole 27 Horizontal plane 16 Huygens principle 11 Inclusion 27 Index of refraction see Refractive index Installation 8 IsGISAXS 6 Island 27 Layer coordinat
6. 13 M Abramowitz and I Stegun Handbook of Mathematical Functions National Bureau of Standards 1964 114 D Babonneau FitGISAXS manual version May 2013 http www pprime fr sites default files pictures d1 FINANO FitGISAXS_130531 zip as per May 2015 115 R W Hendricks J Schelten and W Schmatz Philos Mag 30 819 1974 74 List of Symbols Defines what is on the left page 7 Defines what is on the right page 7 Equal as result of a definition page 7 Asymptotically equal equal in an implied limit page 7 Equal up to first order of a power law expansion hence a special case of asymptotic equality page 7 Matrix element defined as a volume integral page 14 Upward or downward propagating page 19 Glancing angle of the detected beam page 17 Glancing angle of the incident beam page 17 Imaginary part of the refractive index page 19 Small parameter in the refractive index n 1 i page 19 Scattering length density page 11 Scattering or absorption cross section page 15 z dependent factor of y r page 17 Angle between the detected beam projected into the sample plane and the x axis page 17 Angle between the incident beam projected into the sample plane and the x axis page 17 Fourier transform of the perturbation potential x r page 14 Perturbative potential for neutrons equal to the scattering length den sity p page 12 Fourier transform of the pe
7. 30 8 w 60 8 90 7 7 7 7 6 6 6 6 5 5 5 5 _4 4 4 4 gt O 3 3 3 Ey 2 2 2 1 1 1 12 3s 4 5 6 7 6 1 2 3 4 5 6 7 amp D ot 2 3 4 5 6 7 B 12 3 4 5 6 7 8 U U oC pC Figure B 4 Normalized intensity F V computed with L 13 nm W 8 nm H 4 2 nm and a 60 for four different angles w of rotation around the z axis References Agrees with the In plane anisotropic pyramid form factor of IsGISAXS 7 Eq 2 40 8 Eq 217 except for different parametrization and for a refactoring of the analytical expression for F q This is not the anisotropic pyramid of FitGISAXS which is a true pyramid with an off center apex 14 99 B 2 Box cuboid Real space geometry Se L Perspective Top view Side view Figure B 5 A rectangular cuboid Syntax and parameters FormFactorBox length width height with the parameters e length of the base L e width of the base W e height H Form factor etc H L W H F LW H exp i5 sinc 25 sinc 25 sinc gt V LWH S LW Examples Figure B 6 Normalized intensity F V computed with L 18 nm W 4 6 nm H 3 nm for four different angles w of rotation around the z axis 36 IF V and References Agrees with Box form factor of IsGISAXS 7 Eq 2 38 8 Eq 214 except for factors 1 2 in the definitions of parameters L W H 37 B 3 Cone circular 2R Perspective Top view Figure B 7 A truncated
8. 8 Eq 221 which has different parametrization and lacks a factor H in F q ov B 13 Pyramid square based Real space geometry Perspective Syntax and parameters FormFactorPyramid length height Figure B 27 A truncated pyramid with a square base with the parameters L Top view alpha e length of one edge of the square base L e height H e alpha angle between the base and a side face a They must fulfill tan CO H lt L 2 Form factor etc Notation C7 Results F a ah Gs h HH 2 f z exp 2z sinc z 58 Side view ILH 4 H S see gs Pe tana 3tan a S L V a 1 1 A ana 1 6 Ltana S L Examples 5 w 0 IF V Figure B 28 Normalized intensity F V computed with L 10 nm H 4 2 nm and a 60 for four different angles w of rotation around the z axis References Corresponds to Pyramid form factor of IsGISAXS 7 Eq 2 31 8 Eq 221 with different parametrization L 2R scrsxaxs With correction of a sign error and with a more compact form of F q 59 B 14 Ripplel sinusoidal Real space geometry W W Perspective Top view Side view Figure B 29 An infinite ripple with a sinusoidal profile Syntax and parameters FormFactorRipplei length width height with the parameters e length L e width W e height H Form factor etc
9. B 16 Tetrahedron B 17 TruncatedCube B 18 TruncatedSphere B 19 TruncatedSpheroid Bibliography List of Symbols Index 30 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 TU 12 74 75 78 Introduction About BornAgain BornAgain is a software package to simulate and fit reflectometry off specular scat tering and grazing incidence small angle scattering GISAS of X rays and neutrons It provides a generic framework for modeling multilayer samples with smooth or rough interfaces and with various types of embedded nanoparticles Support for neutron polarization and magnetic scattering is under development The name BornAgain alludes to the central role of the distorted wave Born approximation DWBA in the physical description of the scattering process BornAgain is being developed by the Scientific Computing Group of the Julich Centre for Neutron Science JCNS at Heinz Maier Leibnitz Zentrum MLZ Garching Germany It is intended to serve experimentalists in analysing all kinds of reflectometry data It is equally aimed at users of MLZ reflectometers 1 2 3 at JCNS in house researchers and at the reflectometry and GISAS community at large It is the main contribution of JCNS to national 4 and international 5 collaborations of large scale facilities for the development of better user software BornAgain is released as free and open source software under the GNU General Public Lice
10. and DWBA matrix element To compute scattering cross sections in DWBA we first need to determine the distorted wavefunctions w r for r inside the sample The following derivation holds for the incoming wave w i as well as for the back traced detected wave w f We consider wave propagation in one layer with constant average refractive index n2 z n7 A vacuum plane wave impinging on a layered structure is at each interface partly reflected partly refracted so that the wavefunction inside a material layer has an upward and a downward propagating component as per 3 10 Each component is a plane wave with a wavevector koi Ew h wiz 4 1 As explained in connection with 3 4 the in plane wavevector kw remains constant across layer interfaces The vertical wavenumber is obtained from 3 5 Ki wt y Ken AR 4 2 21 layer 0 air vacuum Z0 Z1 layer 1 Z2 ZN 1 layer N 1 ZN layer N substrate Figure 4 1 The parameter z is the z coordinate of the top interface of layer l except for 29 which is the coordinate of the bottom interface of the air vacuum layer 0 We factorize the corresponding wavefunctions as vr e LS d 2 4 3 wl with vertical propagation described by a one dimensional wavefunction A ey 4 4 wl For later convenience the phase factor in 4 4 includes an offset gt as defined in Fig 4 1 The amplitudes A are often written with distinct letters T and R to desi
11. arin Gas 4 21 0 by J Aha iant fiat fi In a scattering setup plane wave amplitudes are subject to two boundary con ditions Let us assume that the source or the sink is located at z gt 0 Then in the top layer Ag 1 is given by the incident or back traced final plane wave In the substrate Aj 0 because there is no radiation coming from z oo This leaves 24 us with two unkown amplitudes the overall coefficients of transmission Aj and re flection Aj These two unknowns are connected by a system of two linear equations Lues M My 4 22 At 0 While it is possible in principle to solve this as a matrix equation the actual imple mentation in BornAgain starts with a unit vector in the substrate and then carries out the propagation step 4 20 interface by interface yielding unnormalized amplitudes Ar anf 4 23 At l 1 N 0 f When the top layer is reached the obtained values are renormalized so that the bound ary condition Ag 1 be satisfied At AF 4 24 AZ For GISAS detection in transmission geometry sink location z lt 0 all the develop ment following 4 21 holds with exchanged order of layers 0 N N 0 At this point it may be an interesting exercise to make a connection with a well known textbook result Consider a system with a single interface between two semi infinite media A straightforward computation will show that the transmission and reflection
12. for certain values of q All these singularities are removable Our implementation comprises appropriate case distinctions Geometrical objects can be parametrized in different ways Concerns about user experience and about code readability sometimes lead to different choices For the Born Again user interfaces GUI and API we have chosen the most standard parameters as used in elementary geometry like length height radius even if this is at variance from the IsGISAXS precedent Where our parametrization made analytic expressions too tedious we use alternate internal parameters to alleviate the formule Examplary form factors are numerically computed in Born approximation The particles are assigned a refractive index of n 107 Parameters are chosen such that the particle volume V is about 250 nm within 5 except ripples which are chosen with a vertical section V L of 40 nm and a length of 25 nm The incident wavelength is 1 A The incident beam is always in x direction hence a 0 Simulated detector images are normalized to the maximum scattering intensity at F 0 V Ias be F a ay pe V B 1 All plots have the same logarithmic color scale extending over ten decades from 107 to 1 Plot ranges in a and o are also standardized as far as reasonably possible For most particle geometries J has horizontal and vertical mirror planes I ae Dp Llar b Jwr Gg Org Ge B 2 For these p
13. whenever n z varies within the sample As a result at any z within the zone where n z varies the vertical wavefunction z is composed of a downward travelling component z and an upward travelling component z For a graded refractive index n that is a smooth function of z the differential equation 3 5 is best solved using the WKB method If otherwise n2 z is discon tinuous at some interface z z then the limiting values of z and z on approaching gt from above or below are connected to each other through Fresnel s transmission and reflection coefficients This applies in particular to multilayer sys tems discussed in chapter 4 3 1 2 Distorted wave Born approximation DWBA The standard form of the Born approximation as presented in Sect 2 2 combines an approximation scheme computing 2 15 by iteration with an assumption the incident field is a plane wave and an analytic result in far field approximation the Green function of the Helmholtz equation is a plane wave with respect to the locus of scattering These three elements must not necessarily go together We can apply the very same approximation scheme even if the incident field is not a plane wave but a distorted wave namely a superposition of downwards and upwards travelling plane waves as derived in the previous section This is the core idea of the distorted wave Born approximation DWBA To carry out this idea we need to determ
14. 2 Neutron scattering in Born approximation 12 2 2 1 The Born expansion zu nn 9 4 nn ea en ne no RS 12 2 2 2 Far field approximation s lt s se e R EEE nn 13 2 2 3 Differential cross section o 2 2 ccm m m ren 15 3 Grazing incidence scattering and the distorted wave Born approxi mation 16 3 1 Scattering under grazing incidence 2 2 2 m En 000 ee eee 16 3 1 1 Wave propagation in 2 1 dimensions 16 3 1 2 Distorted wave Born approximation DWBA 18 Oo AWO get en dee ae Gee un se bwkie ee eee eee 19 4 DWBA for multilayer systems 21 GT palir Clee 2262 tbe Soe eee eRe KEELE EDO SE eS 21 4 1 1 Wave propagation and DWBA matrix element 21 4 1 2 Wave propagation across layers 2 2 2 En nen 23 4 1 3 Damped waves in absorbing media or under total reflection 26 5 Particle Assemblies 27 5 1 Embedded particles a feed ee a a da 0 a a ar Ge deb dea ea 27 A Some proofs 28 A 1 Source detector reciprocity for scalar waves 2 2 2 2 2222 nen 28 B Form factor library B 1 AnisoPyramid rectangle based 00002 ee B 2 Box cuboid B 3 Cone circular B 4 Cone6 hexagonal B 5 Cuboctahedron B 6 Cylinder B 7 EllipsoidalCylinder B 8 FullSphere B 9 HemiEllipsoid B 10 FullSpheroid B 11 Prism3 triangular B 12 Prism6 hexagonal B 13 Pyramid square based 000 eee ee B 14 Ripplel sinusoidal B 15 Ripple2 saw tooth
15. BornAgain Software for simulating and fitting X ray and neutron small angle scattering at grazing incidence User Manual Version 1 3 0 July 31 2015 C line Durniak Marina Ganeva Gennady Pospelov Walter Van Herck Joachim Wuttke Scientific Computing Group J lich Centre for Neutron Science at Heinz Maier Leibnitz Zentrum Garching Forschungszentrum Julich GmbH Homepage Copyright Licenses Authors Disclaimer http www bornagainproject org Forschungszentrum Julich GmbH 2013 2015 Software GNU General Public License version 3 or higher Documentation Creative Commons CC BY SA C line Durniak Marina Ganeva Gennady Pospelov Walter Van Herck Joachim Wuttke Scientific Computing Group at Heinz Maier Leibnitz Zentrum MLZ Garching Software and documentation are work in progress We cannot guarantee correctness and accuracy If in doubt contact us for assistance or scientific collaboration Contents Introduction 5 About Ben Sn 2 0 un 0 nun AD ERR ECL EY ERE Oe 5 About this Mea so amp amp oo 0 2 ee BO wee ee he 6 Typesetting COUVCINIONS u ann a ee a RL RE Ein a 6 1 Online documentation 8 1 1 Download and installation 2 s o a A adri wee EES 8 12 Purther oniliie at Se oro seiss sa prad ee a aa 9 1 3 Registration contact discussion forum 0004 9 2 Small angle scattering and the Born approximation 10 2 1 Coherent neutron propagation 2 a ee 10 2
16. Cone6 Pyramid Cuboctahedron Prism3 Tetrahedron EllipsoidalCylinder Box HemiEllipsoid AnisoPyramid Ripplel Ripple2 COV COV COV On Coy Coy l Ra Ra H L W H a L W H L W H d Page 70 Page 72 Page 38 Page 66 Page 56 Page 40 Page 58 Page 42 Page 54 Page 64 Page 46 Page 36 Page 50 Page 34 Page 60 Page 62 0 20 o 10 10 10 10 10 10 10 10 10 10 10 3 4 5 9 iC IF V KA K 1 2 Figure B 1 Normalized intensity I amp for small angle scattering by a truncated sphere with R 4 2 nm and H 6 1 nm for four different tilt angles V rotation around the y axis Since I possess the standard symmetry B 2 data are only shown for first quadrant O lt hy ap lt D In the following subsections information about the implemented geometries is given in standardized form Analytical expressions are given for the form factor F q for the volume V F 0 and for the maximum horizontal section S the area of the particle as seen from above Mathematical notation in the form factor expressions includes the cardinal sine functions sinc z sin z z and the Bessel function of first kind and first order J z 13 Ch 9 If results contain an integral then no analytical form was found and the integral is evaluated by numeric quadrature The analytical expressions for F q contain singularities
17. Normalized intensity F V computed with R 6 3 nm R 4 2 nm and H 3 nm for four different angles w of rotation around the z axis References Agrees with the IsGISAXS form factor Ellipsoid 7 Eq 2 41 wrongly labeled in Fig 2 4 or Ellipsoidal Cylinder 8 Eq 224 47 B 8 FullSphere Real space geometry 2R 2R Perspective Top view Side view Figure B 17 A full sphere Syntax and parameters FormFactorFullSphere radius with the parameter e radius R Form factor etc sin gR qRcos qR F 4r R exp iq R l ga qh 48 Example Aal VY 5 4 i pe 1 E 0i 2 3 4 5 Pr Figure B 18 Normalized intensity F V computed with R 3 9 nm References This form factor which certainly goes back at least to Lord Rayleigh agrees with the Full sphere of IsGISAXS 7 Eq 2 36 8 Eq 226 AQ B 9 HemiFllipsoid y A Z 2rb X H X 2ra Perspective Top view Real space geometry Side view Figure B 19 An horizontally oriented ellipsoid truncated at the central plane Syntax and parameters FormFactorHemiEllipsoid radius_a radius_b height with the parameters e radius_a in x direction Ra e radius_b in y direction Ry e height equal to radius in z direction H Form factor etc Notation 2 mE Ta z Pr L bz Ry L gt Ve Char a CRT ee Results H J P 2r ee 0 zZ 2
18. a 2 1 dimen sional structure that are on average translationally invariant in x and y direction but structured in z direction By convention we designate the sample plane xy as hori zontal and the sample normal z as vertical even if this does not correspond to the actual experimental geometry The z axis points upwards hence out of the sample towards the vacuum or air halfspace where the incident radiation comes from as illustrated in Fig 3 1 Vertical modulations of the refractive index n r cause refraction and reflection of an incident plane wave For small glancing angles these distortions can be arbitrary large up to the limiting case of total reflection even though 1 n is only of the order 107 or smaller Such zeroth order effects cannot be accounted for by perturbative scattering theory Instead we need to deal with refraction and reflection at the level of the wave propagation equation We move the vertical variations of the squared refractive index to the left hand side of the Schr dinger equation 2 8 V K n z bw r Anx r iblr 3 1 where the overline indicates an horizontal average Deviating from 2 11 the pertur bation has been redefined as u n 2 n r 3 2 In many reflectometers the scattering plane and the sample normal are horizontal in laboratory space X T 16 Layer air Layer Layer Layer Laver Figure 3 1 Geometric conventions in GISAS scatt
19. ahedron removed from the cube s vertices t They must fulfill E Form factor etc Notation Besides the form factor Fpox q of the full cube of side length L Sect B 2 we need the form factor of a trirectangular tetrahedrons as cut from the cube Pae L T Lb a Gene L LC 0 i ee x4 sinc exp 3 sinc dy 2 dy iy Tia 2 2 1 L L Cd exp 2 sinc dy T qz 2 2 66 Thanks to symmetry see the following figure which shows the vertices V for i 1 8 the form factors of other seven tetrahedrons cut from the cube can be com puted as follows note that the origin is taken as usual at the centre of the bottom face of the cube KS tar dys Fa Wz L t D ertex as ly 42 L t Pyertex a dx 4z L t vertex et vertex LH 0 0 ore z3 A E 4x re Wey Vestal vertex Ves Vy Iz Dit Ge de a exp q L Frertex CE dy zs L t Perle ax y H lt L t exp iq L Frertex dys He Tiz L t L Hr ly H lt L t exp iq L Frertex dz ly Az L t cries ax dys H lt L t exp q L Frertex a lx dz L t Result 8 i pal ly H lt L L L 7 gt P vertex Cor Cy Wz L t c 1 V LB S L Examples 67 Figure B 36 Normalized intensity F V computed with L 25 nm W 10 nm H 8 nm and d 5 nm for four different angles w of rotation around the z axis Refer
20. articles plots of J are restricted to the quadrant a gt 0 amp gt 0 However it requires some experience to fully appreciate the information content of these plots For 32 Hai V O V 20 10 107 107 10 107 107 10 10 10 10 10 4 2 0 2 4 9 Figure B 2 Same data as in Fig B 1 but now shown for all four quadrants 5 lt dp ag lt 5 The vertical interference pattern which gradually disappears with increasing tilt angle is much more salient in this plot than in the preceding one quadrant representation a demonstration of this try to capture the main features of Fig B 1 Then compare with Fig B 2 33 B 1 AnisoPyramid rectangle based Real space geometry rn Perspective Top view Side view m lt x x x Figure B 3 A truncated pyramid with a rectangular base Syntax and parameters FormFactorAnisoPyramid length width height alpha with the parameters e length of the base L e width of the base W e height H e alpha angle between the base and a side face a They must fulfill tan a tan H lt L and H lt W Form factor etc Notation t L 2 w W 2 h H 2 f z exp tiz sinc z Results P rp Es CRT ta 8 easta ae ea DE t Farat tt P tant aye f S lt a a enra ganh 34 L W H 4 H Y HIL SI 4 2 tan q 3 tan a 5 LW Examples 8 w 0 8 w
21. assical approximation see WKB method Shape transform 73 Sign convention 20 scattering vector 14 wave propagation 10 Sinusoidal ripple form factor 60 Small angle scattering 10 12 Snell s law 22 Sphere form factor 48 truncated 70 Spheroid form factor 52 truncated 72 Tetrahedron form factor 64 Total reflection 26 Transfer matrix 24 Transition matrix 14 Transmission see Fresnel coefficients Transmission geometry 25 Truncated cone form factor 38 Truncated ellipsoid form factor 50 Truncated pyramid form factor hexagonal Cone6 40 rectangular AnisoPyramid 34 square 58 Truncated sphere form factor 70 Truncated spheroid form factor 72 Truncated tetrahedron form factor 64 Tunneling 26 80 Tutorials 9 Vertical direction 16 Wave propagation see also Sign convention coherent 11 neutrons 10 12 neutrons polarized 6 X rays 6 Wavevector complex 26 Windows see Microsoft Windows WKB method 18 X rays propagation and scattering 6
22. cone with circular base Real space geometry Syntax and parameters FormFactorCone radius height alpha with the parameters e radius R e height H e alpha angle between the side and the base a They must fulfill H lt Rtana Form factor etc Notation H Rg R BIS VRE t f d tana Results gt Ji qye _ R F 2r tan g eiin f dp pf e Ry IP g 3 3 S 7R 38 Side view Examples Figure B 8 Normalized intensity F V computed with R 4 nm H 11 nm a 75 for four different tilt angles 0 rotation around the y axis References Ha V and Agrees with Cone form factor of IsGISAXS 7 Eq 2 28 8 Eq 225 except for a substitution z p in our expression for F 39 B 4 Cone6 hexagonal Real space geometry a lt gt ih Perspective Top view Side view Figure B 9 A truncated hexagonal pyramid Syntax and parameters FormFactorCone6 radius height alpha with the parameters e radius of the regular hexagonal base R e height H e alpha between the base and a side face a Note that the orthographic projection does not show a but the angle 6 between the base and a side edge They are related through 3tana 2tan 8 The following is written more conveniently in terms of 6 The parameters must fulfill H lt tan B R Form factor etc Notation H 1 _ v3 _ Aura Tahe Gy Pay a tan
23. dge L e height H Form factor etc 2 3 L L f 1q lt aga P iaz ex LST 08 4055 C Vay sine 1 5 H H x H sinc a3 exp i5 s v3 ein 4 7 R A 54 Examples 5 LU Ha V o 30 10 10 107 10 104 10 10 107 10 10 10 3 4 5 C 1 2 Figure B 24 Normalized intensity F V7 computed with R 13 8 nm and H 3 nm for four different angles w of rotation around the z axis References Agrees with Prism3 form factor of IsGISAXS 7 Eq 2 29 8 Eq 219 except for the definition of parameter L 2Ryocrsayg In FitGISAXS just called Prism 14 59 B 12 Prism6 hexagonal Real space geometry I Perspective Top view Side view Figure B 25 A prism based on a regular hexagon Syntax and parameters FormFactorPrism6 radius height with the parameters e radius of the hexagonal base R e height H Form factor etc AHV3 H H a3 HE a2 exp i5 x F d h 3q R R l a sinc 25 sinc E cos q R cos a cos 2 l 3V3 2 V HR 2 Examples 56 w 30 Fa V O 10 10 107 10 107 107 10 10 10 10 10 x 3 4 5 0 1 2 pC Figure B 26 Normalized intensity F V computed with R 5 7 nm and H 3 nm for four different angles w of rotation around the z axis References Corresponds to Prism6 form factor of IsGISAXS 7 Eq 2 31
24. dge of the equilateral triangular base L e height H e alpha angle between the base and a side face a They must fulfill tana H lt L 2 3 Note that the orthographic projection does not show a but the angle 6 between the base and a side edge They are related through tana 2 tan Form factor etc Notation B 1 de V3 qy 1 tV tay A a B Ltana 1757 tan a z 42 5 tan a sl 43 tana 2 y3 Results za __V 3H E Ta CA Dr oh da 13 39 2 3 24 bi tda D sinc q3H qs V3q expliq D sine q H dz V3qy exp iq2 D sine q2H IL 64 s 872 Examples w 0 Hai V Figure B 34 Normalized intensity F V computed with L 12 nm H 8 nm and a 75 for four different angles w of rotation around the z axis The low symmetry requires other angular ranges than used in most other figures References Agrees with the Tetrahedron form factor of IsGISAXS 7 Eq 2 30 8 Eq 220 In FitGISAXS correctly called Truncated tetrahedron 14 65 B 17 TruncatedCube Real space geometry L L Perspective Top view Side view Figure B 35 A cube whose eight vertices have been removed The truncated part of each vertex is a trirectangular tetrahedron Syntax and parameters FormFactorTruncatedCube length removed_length with the parameters e length of the full cube L e removed_length side length of the trirectangular tetr
25. e 22 index 21 22 transfer matrix 24 Layer structures see Multilayer Lazzari Remi 6 Linux 8 Lippmann Schwinger equation 13 MacOS 8 Mesoparticles see Particles Microsoft Windows 8 Momentum transfer see Scattering vector Multilayer 21 26 coordinates 22 numbering 21 22 transfer matrix 24 Nanoparticles see Particles Neutrons polarization 6 wave propagation 10 12 Newsletter 9 Operating system 8 Optical potential Fourier transform 14 macroscopic 11 nuclear microscopic 11 Particle assemblies 6 27 Perturbation 12 Phase integral method see WKB method Platform operating system 8 Polarized neutron 79 propagation and magnetic scattering 6 Potential see Optical potential see Perturbation Prism form factor hexagonal Prism6 56 reactangular Box 36 triangular Prism3 54 Pyramid form factor hexagonal Cone6 40 rectangular AnisoPyramid 34 square 58 Python 9 Quadrature 32 Reciprocity 18 28 29 Reflection 16 see also Fresnel coefficients Refraction 16 Snell s law 22 Refractive index 11 graded 18 sign convention 11 19 Registration 9 Ripple form factor saw tooth Ripple2 62 sinusoidal Ripplel 60 Roughness 6 Sample normal 16 Sample plane 16 SAS see Small angle scattering Saw tooth ripple form factor 62 Scattering length density 11 Scattering vector 14 Schrodinger equation macroscopic 11 microscopic 10 Semicl
26. e obtain elkr a Gurt ar r 3 9 which agrees literally with 2 20 though is not any longer a plane wave Accord ingly the scattered far field is still given by 2 22 and the differential cross section by 2 31 We only need to redetermine the matrix element ab x 2b which no longer has the plane wave form 2 25 Since both the incident and the scattered distorted wavefunction are composed of downward and upward propagating waves wi Tr yuri yar with w f 3 10 the matrix element can be expanded into four terms L INTO Wr lixe rixen WE alba BR 3 11 or in an obvious shorthand notation blxlbe S a 312 a This equation contains the essence of the distorted wave Born approximation for small angle scattering under grazing incidence and is the base for all scattering models implemented in BornAgain Since w v w appears as a squared modulus in the differential cross section 2 31 the four terms of 3 12 can interfere with each other which adds to the complexity of GISAS patterns 3 2 Absorption The complex refractive index of a given material shall be written as n 1 6 i8 3 13 introducing two small real parameters 6 However in our derivations which are all rooted in 2 8 n only appears as n Therefore we actually define n 1 26 258 3 14 and read 3 13 as an excellent approximation While the real part of n is responsible for refraction ref
27. ences 15 68 10 107 107 107 107 107 10 10 10 10 10 IF V Page intentionally left blank 69 B 18 TruncatedSphere Real space geometry 2R Perspective Top view Side view Figure B 37 A truncated sphere Syntax and parameters FormFactorTruncatedSphere radius height with the parameters e radius R e height H They must fulfill U lt H lt 2R Form factor etc Notation di gt deta R VRP 2 Results F 2rexpliq H R J dz R2 expliq z dz R H ojia 2 H R 1 H R van eG 4 E TR TL TR H gt R n 2RH H H lt R 70 Figure B 38 Normalized intensity F V7 computed with R 4 2 nm and H 6 1 for four different tilt angles 7 rotation around the y axis References HAP V nm Agrees with the IsGISAXS form factor Sphere 7 Eq 2 33 or Truncated sphere 8 Eq 228 71 B 19 TruncatedSpheroid Real space geometry 2R Perspective Top view Side view Figure B 39 A vertically oriented horizontally truncated spheroid Syntax and parameters FormFactorTruncatedSpheroid radius height height_flattening with the parameters e radius R e height H e height_flattening fp They must fulfill H U lt R 2 Form factor etc Notation q 0 R v I e Results Ty kt gt J q R F 2rexplig H fp P dz RE _ exp iq z f R H qiz H
28. eory exposed in this manual is actually used in BornAgain Such a box contains an important fact for instance an equation that has a central role in the further development of the theory Variations of the equation sign as are explained in the symbol index page 75 See there as well for less common mathematical functions like the cardinal sine function sinc Chapter 1 Online documentation This User Manual is complementary to the online documentation at the project web site http www bornagainproject org It does not duplicate information that is more conveniently read online This brief chapter contains no more than a few pointers to the web site BornAgain m g Simulate and fit grazing incidence small angle scattering Home DOCE Documentation Contact Forums About LATEST RELEASE BornAgain 0 9 9 2014 10 29 SEARCH Q m Simulate your GISAS experiment USER MENU M t Welcome to BornAgain s My accoun o Log out BornAgain is a free software package to simulate and fit small angle scattering at grazing incidence It supports analysis of both X ray GISAXS and neutron GISANS data Its name BornAgain indicates the central role of the distorted wave Born approximation in the physical description of the scattering process The software provides a generic framework for modeling multilayer samples with smooth or rough interfaces and with various types of embedded nanoparticles Read more Fi
29. ering comprise a Cartesian coordinate system and a set of angles The coordinate system has a z axis normal to the sample plane and pointing into the halfspace where the beam comes from The x axis usually points along the incident beam projected onto the sample plane Incident and final plane waves are characterized by wavevectors k k the angle a is the incident glancing angle is usually zero unless used to describe a sample rotation a is the exit angle with respect to the sample s surface and D is the scattering angle with respect to the scattering plane The numbered layers illustrate a multilayer system as dicussed in Sect 4 which only accounts for horizontal fluctuations of the refractive index Wave prop agation unperturbed by x but including refraction and reflection effects obeys the homogeneous equation V2 K n 2 y r 0 3 3 It is solved for the horizontal coordinate ru by the factorization ansatz nr eid 2 3 4 The horizontal wavevector kyj remains constant as initialized by the incoming beam The vertical wavefunction must fulfill 02 K n2 z k LAL 20 3 5 k z When an incident plane wave travelling downwards with z e impinges on a sample with n2 z 1 then the wave is partly reflected k reversed into 17 k and partly refracted k changing while k stays constant resulting in a change of glancing angle Similarly reflection and refraction occur
30. eter that characterizes the incoming radiation In terms of K and n the macroscopic Schr dinger equation 2 4 can be rewritten as 11 V2 K2n r 2 u r 0 2 8 This equation is the starting point for the analysis of all small angle scattering exper iments whether under grazing incidence GISAS or not regular SAS 2 2 Neutron scattering in Born approximation 2 2 1 The Born expansion To describe an elastic scattering experiment we need to solve the Schrodinger equa tion 2 8 under the asymptotic boundary condition Kr Arr w r v r f p for r gt 2 9 where w r is the incident wave as prepared by the experimental apparatus and the second term on the right hand side is the outgoing scattered wave that carries information in form of the angular distribution f V y For thermal or cold neutrons as for X rays the refractive index n is almost always very close to 1 This suggests a solution of the Schrodinger equation by means of a perturbation expansion in powers of n 1 This expansion is named after Max Born who introduced it in quantum mechanics To carry out this idea we rewrite the Schrodinger equation once more so that it takes the form of a Helmholtz equation with a perturbation term on the right side V K g r dey n n 2 10 with x n J IL i 2 11 This definition just compensates 2 6 so that Y p In the following we prefer the notation Y and the appellation perturba
31. gnate the transmitted or reflected beam wW wl At 4 5 wl 7 They need to be computed recursively as described in the following subsection 4 1 2 In the absence of absorption wavevectors are real so that we can describe the beam in terms of a glancing angle G arctan k kiap 4 6 wl Lwl w Equivalently bia Kn cos awyr 4 7 Since kw is constant across layers we have N COS a the same for all J 4 8 which is Snell s refraction law Since the w are plane waves within layer l we can at once write down the DWBA transition matrix element 3 12 DilxlWp 22 Alt LX k ki 4 9 22 where Aa rE J d r etar y r 4 10 Zi is the Fourier transform of the perturbative potential 3 2 restricted to one layer To alleviate later calculations we now number the four DWBA terms from 1 to 4 and define the corresponding wavenumbers and amplitude factors and as dk ake Oris ATA 2 k kt 02 AAT S f 4 11 q ke k C AF A g ek k C Ra E Accordingly we can write 4 9 as dlxldr gt C C xT 4 12 l U From 4 1 we see that all four wavevectors q have the same horizontal component q q as Qi z 4 13 whence the vertical components qi k kij 2 q k k iL fi L 4 14 q Ke kils gi be ki 4 1 2 Wave propagation across layers The plane wave amplitudes A need to be computed recursively from layer to
32. gure 1 1 A screenshot of the home page http www bornagainproject org 1 1 Download and installation BornAgain is a multi platform software We actively support the operating systems Linux MacOS and Microsoft Windows The DOWNLOAD section on the BornAgain web site points to the download location for binary and source packages It also provides a link to our git server where the unstable development trunk is available for contributors or for users who want to live on the edge The DOCUMENTATION section contains pages with Installation instructions 1 2 Further online information The DOCUMENTATION section of the project web site contains in particular e an overview of the software architecture e a list of implemented functionality e tutorials for Working with BornAgain using either the Graphical User Inter face or Python scripts e a comprehensive collection of examples that demonstrate how to use BornAgain for modeling various sample structures and different experimental conditions e a link to the API reference for using BornAgain through Python scripts or C programs 1 3 Registration contact discussion forum To stay informed about the ongoing development of BornAgain register on the project homepage http www bornagainproject org Create new account You will then receive our occasional newsletters and be authorized to post to the discussion forum To contact the BornAgain development and maintenance tea
33. in Eqs 4 6 4 8 is untenable Writing for a decomposition into a real and an imaginary part we find an exponential decay of the plane wave amplitudes b 2 oF RL 4 27 along their propagation direction z With an analogous decomposition of the three dimensional wavevector 4 1 we obtain for the flux defined as in 2 27 eh 4 28 In the special case of a pure imaginary k the flux direction is k ky Then y r is an evanescent wave travelling horizontally Since a stationary evanescent wave implies that there is no vertical energy transport all incoming radiation undergoes total reflection In the generic case of a complex k the flux has a vertical component Accord ingly the total reflection is not perfect Some intensity is dissipated in layer l And if layer L lt N is not too thick then some radiation intensity also tunnels into the adjacent layer l 1 26 Chapter 5 Particle Assemblies 5 1 Embedded particles In many important GISAS applications fluctuations of the refractive index are due to islands inclusions or holes of a mesoscopic size nanometer to micrometer In the following all such inhomogeneities will be described as particles that are embedded in a material layer Implemented particle form factors are described in Appendix B 27 Appendix A Some proofs This appendix contains proofs that were taken out of the main text in order not to disrupt the physics narratio
34. ine the Green function G In Sect 2 2 1 we did so quite specifically for a homogeneous material Computing G in closed form for a more generic wave equation like 3 3 is far more difficult if not outright impossible Fortunately this computation is not necessary and would be but wasted effort We do not need the full solution G r r but only its asymptotic far field value G rp r at a detector position rp Thanks to a source detector reciprocity theorem A 10 proven in Appendix A 1 we can compute this value as G rp r Bir rp 3 6 where B is the adjoint Green function that describes backward propagation from rp into the sample Outside the sample B obeys the Helmholtz equation with isolated inhomogeneity 2 13 and therefore has the far field expansion 2 20 L K Tn S mu 3 7 Bey TP rp Arr py When this backward propagating plane waves impinges on the sample it undergoes reflection and refraction in exactly the same way as the incident plane wave e 7 Also called semiclassical approximation or phase integral method named after Wentzel 1926 Kramers 1926 Brillouin 1926 See any textbook on quantum mechanics 3The distorted wave Born approximation was originally devised by Massey and Mott ca 1933 for collisions of charged particles 18 Therefore 3 7 admits a generalization that also holds inside the sample etkrp DLP rp ee 3 8 Applying now the reciprocity theorem 3 6 w
35. jugate of w With respect to r Lx is an outgoing spherical wave The scattered wave 2 17 becomes in the far field approximation Kr Ps far T ex 7 2 22 T where we used Dirac notation for the transition matrix element xl f Prex 2 23 In order to reconcile conflicting sign conventions we will in the following rather use its complex conjugate Yel Y TD wely w Under the standard assumption that the incident radiation is a plane wave fr et 2 24 with k K the matrix element takes the form WIT dre tiny r etten d retarx r x q 2 25 where we have introduced the scattering vector q r k k 2 26 and the notation x q for the Fourier transform of the perturbative potential which is what small angle neutron scattering basically measures 3With this choice of sign hg is the momentum gained by the scattered neutron and lost by the sample In much of the literature the opposite convention is prefered since it emphasizes the sample physics over the scattering experiment However when working with twodimensional detectors it is highly desirable to express pixel coordinates and scattering vector components with respect to equally oriented coordinate axes which can only be achieved by the convention 2 26 14 2 2 3 Differential cross section In connection with 2 16 we mentioned that a scattering experiment measures inten sities r We shall now restate this in a
36. l in upward reflection direction also denoted A page 22 wl Subscript scattered page 13 Cardinal sine sinc x sin x x page 7 Maximum horizontal section of embedded particle page 32 Time page 10 Partial amplitude of w r in layer l in downward transmission direc tion also denoted A page 22 wl Macrosopic optical potential page 11 Microscopic optical potential page 11 An index that can take the values i incident or f final page 19 Horizontal coordinate usually chosen along the incoming beam projec tion page 17 Horizontal oordinate chosen normal to z and x page 17 Vertical coordinate at the top of layer l at the bottom for 0 page 22 Vertical coordinate along the sample normal page 16 Unit vector along the sample normal page 21 17 Index Absorption 19 20 Anisotropic pyramid form factor 34 API see Application programming interface Application programming interface 9 Born approximation 12 13 Box form factor 36 Bragg scattering by atomic lattices 11 Bug reports 6 C 9 Citation 5 Coherent forward scattering 11 Coherent wavefunction 11 Cone form factor circular 38 hexagonal Cone6 40 Continuum approximation neutron propagation 11 Conventions see Sign convention see Horizontal and Vertical Coordinate system 14 Cross section 15 Cube form factor facetted 66 Cuboctahedron form factor 42 Cuboid form factor 36 Cylinder
37. layer Since these computations are identical for incident and final waves we omit the sub script w in the remainder of this section At layer interfaces the optical potential changes discontinuously From elementary quantum mechanics we know that piecewise solutions of the Schrodinger equations must be connected such that the wavefunction o r and its first derivative V r evolve continuously To deal with the coordinate offsets introduced in 4 4 we introduce the function dy 2 241 4 15 which is the thickness of layer l except for L 0 where the special definition of zo Fig 4 1 implies d 0 We consider the interface between layers l and 1 23 Mia Zj 1 layer 1 1 _1 di 1 Mi Z layer dD dj Mit ZIL Figure 4 2 The transfer matrix M connects the wavefunctions _ in adjacent layers with L 1 N as shown in Fig 4 2 This interface has the vertical coordinate 2 2 _1 d _ Accordingly the continuity conditions at the interface are ilz b1 1 21 1 din 4 16 0 01 2 0 01 1 41_1 di1 We abbreviate fi ku K yn hy K 4 17 and jp Se ee 4 18 For the plane waves 4 4 the continuity conditions 4 16 take the form A 47 47 16 1 LAT 191 A f Atf A At 5 en IEA fi 0 TA 0 TR After some lines of linear algebra we can rewrite this equation system as Ary A Vale aa with the transfer matrix TEN dj U 1
38. lection and scattering the imaginary part describes absorption and leads to a damping of propagating waves The 19 plus sign in front of the imaginary part is a consequence of the quantum mechanical sign convention in the X ray crystallography convention it would be a minus sign The factorization ansatz 3 4 leaves us some freedom how to deal with an imag inary part of n We choose that horizontal wavevectors ky shall always be real The damping then appears in the vertical wavefunction z that is governed by the com plex wave equation 3 5 20 Chapter 4 DWBA for multilayer systems In Sect 3 1 we have discussed wave propagation and scattering in 2 1 dimensional systems that are translationally invariant in the horizontal ry plane and have a vertical refractive index profile n z Here we specialize to layered systems where n z is a step function that is constant within one layer First only scalar interactions are considered Later the theory is extended to account for polarization effects By convention layers are numbered from top to bottom see Fig 4 1 The top vacuum or air layer which extends to z 00 has number 0 the substrate extending to z oo is layer N All layer interfaces are assumed to be perfectly smooth Support for rough inter faces is already implemented in BornAgain but documentation is adjourned to a later edition of this manual 4 1 Scalar case 4 1 1 Wave propagation
39. m in the Scientific Computing Group of Heinz Maier Leibnitz Zentrum MLZ Garching write a mail to contact bornagainproject org or fill the form in the CONTACT section of the project web site For questions that might be of wider interest please consider posting to the discussion forum accessible through the FORUMS tab of the project web site Chapter 2 Small angle scattering and the Born approximation This chapter introduces the basic theory of small angle scattering SAS We specifi cally consider scalar neutron propagation adjourning the notationally more involved vectorial theory of X rays and polarized neutrons a later edition Our exposition is self contained except for the initial passage from the microscopic to the macroscopic Schr dinger equation which we outline only briefly Sect 2 1 The standard descrip tion of scattering in first order Born approximation is introduced in a way that is suitable subsequent modification into the distorted wave Born approximation needed for grazing incidence small angle scattering Sect 2 2 2 1 Coherent neutron propagation The scalar wavefunction r t of a free neutron is governed by the microscopic Schr dinger equation iho y r t e vir wr t 2 1 By assuming a time independent potential V r we have excluded inelastic scattering Therefore we only need to consider monochromatic waves with given frequency w In consequence we have a stationary wavef
40. more rigorous way In the case of neutron scattering one actually measures a probability flux We define it in arbitrary relative units as Y Y Tir ZU WT 2 27 The ratio of the scattered flux hitting an infinitesimal detector area r7dQ to the incident flux is expressed as a differential cross section do r J r AN J 2 28 With 2 24 the incident flux is With 2 22 the scattered flux at the detector is K Ir F le 2 30 From 2 28 we obtain the generic differential cross section of elastic scattering in first order Born approximation do CE 2 31 As we shall see below it holds not only for plane waves governed by the vacuum Helmholtz equation 2 12 but also for distorted waves In the plane wave case 2 25 considered here the differential cross section is just the squared modulus of the Fourier transform of the perturbative potential Z xg 2 32 15 Chapter 3 Grazing incidence scattering and the distorted wave Born approximation In this chapter introduce grazing incidence small angle scattering and its standard theoretical treatment by means of the distorted wave Born approximation Sect 3 1 We also discuss the treatment of absorption Sect 3 2 3 1 Scattering under grazing incidence 3 1 1 Wave propagation in 2 1 dimensions Reflectometry and grazing incidence scattering are designed for the investigation of surfaces interfaces and thin layers or most generically samples with
41. n A l Source detector reciprocity for scalar waves We derive a source detector reciprocity theorem for the scalar Schrodinger equation It is needed in the derivation of the distorted wave Born approximation Sect 3 1 2 where it allows us to short cut the computation of the Green function yielding at once the far field at the detector position We start from a generic stationary Schrodinger equation with an isolated inho mogeneity V2 v r G r rg 6 r rg A 1 We assume that the source location rg which in our application is a scattering center lies within a finite sample volume Outside the sample the potential v r has the constant value K so that A 1 reduces to the Helmholtz equation V K7 G r rg 0 A 2 We introduce the adjoint Green function B that originates from a source term at the detector location and obeys IV ov r B r rp 6 r rp A 3 We also introduce the auxiliary vector field X r Ts rp Bir rp VG r rg G r rg VB r rp A 4 We inscribe the sample the detector and the origin of the coordinate system into a sphere 5 with radius R and compute the volume integral I rg rp arvxc Pa Tn S ar BV G GV7B S B rg rp E G rp Tg 28 Alternatively we can compute J as a surface integral I rs rp J do X r rs rp J do B RG G gB A 6 OS OS On the surface 05 B and G are outgoing wave fields that obey the Helmholtz equation
42. nse GPL version 3 or higher This documentation comes under the Creative Commons license CC BY SA The software BornAgain embodies nontrivial scientific ideas Therefore when BornAgain is used in preparing scientific papers it is mandatory to cite the software C Durniak M Ganeva G Pospelov W Van Herck J Wuttke 2015 BornAgain Software for simulating and fitting X ray and neutron small angle scattering at grazing incidence version http www bornagainproject org The initial design of BornAgain owes much to the widely used program IsGISAXS by R mi Lazzari 6 7 Therefore when using BornAgain in scientific work it might be appropriate to also cite the pioneering papers by Lazzari et al 6 8 Since version 1 0 BornAgain almost completely reproduces the functionality of IsGISAXS About 20 exemplary simulations have been tested against IsGISAXS and found to agree up to almost the last floating point digit BornAgain goes beyond IsGISAXS in supporting an unrestricted number of layers and particles diffuse re flection from rough layer interfaces and particles with inner structures Support for neutron polarization and magnetic scattering is under development Adhering to a strict object oriented design BornAgain provides a solid base for future extensions in response to specific user needs About this Manual This user manual is complementary to the online documentation at http www bornagainprojec
43. probabilities determined as above agree with Fresnel s result for s polarized light 2 2 Jom Fay 4 25 1p _ 2ho An ln T fi Motf 2 MAT 7 The above algorithm fails if f 0 because M becomes singular A layer with f 0 only sustains horizontal wave propagation radiation from below or above is totally reflected at its boundaries In BornAgain such total reflection is imposed if f falls below a very small value currently 10 7 However except for the top vacuum layer this ought to be inconsequential because the index of refraction should always have an absorptive component that prevents f from becoming zero See any optics textbook e g Born amp Wolf 11 ch 1 5 2 or Hecht 12 ch 4 6 2 29 4 1 3 Damped waves in absorbing media or under total reflection In Sect 3 2 we have chosen the horizontal wavevector k to be always real and constant In contrast the vertical wavenumber k given by 4 2 can become imaginary or complex If n is real and smaller than nj cos ag then Snell s law of refraction 4 8 cannot be fulfilled and the radicand in 4 2 becomes negative so that k becomes pure imaginary If the layer is absorbing described by a positive imaginary part of n2 then the radicand in 4 2 becomes complex and the wavenumber k as well For complex k the theory developed above remains applicable except that the geometric interpretation of the wavevectors k
44. ring by atomic lattices only occurs at angles far above the small angle range covered in GISAS experiments Accordingly it can be neglected in the analysis of GISAS data or at most is taken into account as a loss channel Therefore we can neglect the atomic structure of V r and perform some coarse eraining to arrive at a continuum approximation This is similar to the passage from the microscopic to the macroscopic Maxwell equations The details are intricate 9 10 but the result 9 eq 2 8 32 looks very simple The macroscopic field equation has still the form of a stationary Schrodinger equation a 1 57 v r hu w r 0 2 4 where 10 now stands for the coherent wavefunction obtained by superposition of incident and forward scattered states and u r is the macroscopic optical potential This potential is weak and slowly varying compared to atomic length scales It can be rewritten in a number of ways especially in terms of a bound scattering length density p r 9 eq 2 8 37 u Ih m u r Ps r 2 5 or of a refractive index n r defined by 2 An 2m n r 1 72Ps r 1 72 2 UI 2 6 In the latter expression we introduced the vacuum wavenumber K which is connected with the frequency w through the dispersion relation h kK TW 2 7 2m Since we only consider stationary solutions 2 2 w will not appear any further in our derivations Instead we use K as the given param
45. rturbation potential x r evaluated in one sample layer page 23 19 wr Wea r WL W ki k wl Stationary wavefunction page 10 Microscopic neutron wavefunction page 10 Coherent wavefunction page 11 Plane wave propagating from the sample towards the detector page 14 Incident wavefunction page 12 Scattered wavefunction page 13 Far field approximation to the scattered wavefunction Y r page 14 Upward or downward propagating component of y r page 19 Frequency of incident radiation page 10 Solid angle page 15 Amplitude of the plane wave LP page 22 Green function adjoint of G page 18 Subscript final for outgoing waves scattered into the direction of the detector page 14 Green function page 13 Far field approximation to the Green function G r r page 14 Subscript incident page 12 Bessel function of first kind and first order page 32 Probability flux page 15 Component of k along the sample normal page 17 wavevector page 14 Projection of k onto the sample plane page 17 wavevector of the plane wave 4 r page 21 Vacuum wavenumber corresponding to the frequency w page 11 Index of layer in multilayer sample page 21 Neutron mass page 11 Refractive index page 11 Refractive index horizontally averaged page 16 76 rD wl sinc Scattering vector page 14 Position page 10 Position of the detector page 18 Partial amplitude of w r in layer
46. s Results Ver ph a ee S _ S F A dapet pg sinc q P sin q p cos 2q p cos q p cos 4 x H V tan 8 R3 R3 3V 3R Se 40 Examples Ha V Figure B 10 Normalized intensity F V computed with R 6 nm H 5 nm and a 60 for four different angles w of rotation around the z axis References Hopefully agrees with Cone6 form factor of IsGISAXS 7 Eq 2 32 8 Eq 222 except for different parametrization Al B 5 Cuboctahedron Real space geometry L Perspective Top view Side view Figure B 11 A compound of two truncated pyramids with a common square base and opposite orientations Syntax and parameters FormFactorCuboctahedron length height height_ratio alpha with the parameters e length of the shared square base L e height of the bottom pyramid H e height_ratio between the top and the bottom pyramid ry e alpha angle between the base and a side face a They must fulfill tan a tan a L and r H lt H lt L Form factor etc Using the form factor of a square pyramid Fp Sect B 13 F exp ig H Fy de y 42 L r gH 0 Pages 4y des L HAH 1 2H 3 Wah AS V t LIS 1 1 E 6 ao Ltan L Lano S L 42 Examples Ha V Figure B 12 Normalized intensity F V7 computed with L 8 nm H 5 nm ry 0 5 and a 60 for four different angles w of rotation around the z a
47. t org It does not duplicate information that is more conveniently read online Therefore Sect 1 just contains a few pointers to the web site The remainder of this manual mostly contains background on the scattering theory and on the sample models implemented in BornAgain and some documentation of the corresponding Python functions AS A This manual is incomplete Several important chapters are still missing Specifically we plan to provide documentation on e X ray propagation and scattering e polarized neutron propagation and magnetic scattering e mapping of 0a OQ onto flat detectors e scattering by rough interfaces e scattering by particle assemblies We intend to publish these chapters successively along with new software release To avoid confusion starting with release 1 2 the manual carries the same version number as the software even though it is in a less mature state We urge users to subscribe to our newsletter see Sect 1 3 and to contact us for any question not answered here or in the online documentation We are grateful for all kind of feedback criticism praise bug reports feature requests or contributed modules If questions go beyond normal user support we will be glad to discuss a scientific collaboration Typesetting conventions In this manual we use the following colored boxes to highlight certain information Such a box contains an implementation note that explains how the th
48. tive potential over the scattering length density p to prepare for the generalization to the electromagnetic case Equation 2 10 looks like an inhomogeneous differential equation provided we neglect for a moment that the unknown function 10 reappears on the right side The homogeneous equation V2 K2 wr 0 2 12 is solved by plane waves and superpositions thereof It applies in particular to the incident wave Tt goes back to Lord Rayleigh who devised it for sound and later also applied it to electromagnetic waves which resulted in his famous explanation of the blue sky 12 For an isolated inhomogeneity V K G r r r r 2 13 is solved by the Green function ei K r r G r r 2 14 Arr r which is an outgoing spherical wave centered at r Convoluting this function with the given inhomogeneity 47y w we obtain what is known as the Lippmann Schwinger equation E J d r G r r jarxlr jlr 2 15 This integral equation for y r improves upon the original stationary Schr dinger equation 2 10 in that it ensures the boundary condition 2 9 It can be resolved into an infinite series by iteratively substituting the full right hand side of 2 15 into the integrand Successive terms in this series contain rising powers of x Since x is assumed to be small the series is likely to converge In first order Born approximation only the linear order in y is retained
49. unction Wr t rle 2 2 The minus sign in the exponent of the phase factor is an inevitable consequence of the standard form of the Schrodinger equation and is therefore called the quantum mechanical sign convention For electromagnetic radiation usage is less uniform While most optics textbooks have adopted the quantum mechanical convention 2 2 in X ray crystallography the conjugate phase factor et is prefered This crystallographic sign convention has also been chosen in influential texts on GISAXS e g 8 Here however we are concerned not only with X rays but also with neutrons and therefore we need to leave the Schr dinger equation 2 1 intact Thence 10 In this manual and in the program code of BornAgain the quantum mechanical sign convention 2 2 is chosen This has implications for the sign of the imaginary part of the refractive index as explained in Sect 3 2 Inserting 2 2 in 2 1 we obtain the stationary Schr dinger equation l V r hu wir 0 2 3 The nuclear or microscopic optical potential V r in a somewhat naive conception 19 p 7 consists of a sum of delta functions representing Fermi s pseudopotential The superposition of the incident wave with the scattered waves originating from each illuminated nucleus results in coherent forward scattering in line with Huygens prin ciple Coherent superposition also leads to Bragg scattering However Bragg scatte
50. us Qp dre are shown for small angle scattering conditions a d 0 The computation of F q is based on shapes S r given in Cartesian coordinates as defined in the orthogonal projections Typically the vertical z direction is chosen along a symmetry axis of the particle The origin is always at the center of the bottom side of the particle Different parametrization or a different choice of the origin cause our analytic form factors to trivially deviate from expressions given in the IsGISAXS manual 7 Sect 2 3 or in the literature 8 Appendix We recomputed all expressions to make sure that they also hold for complex scat tering vectors used to describe in order to take any material absorption into account The implementation in BornAgain allows all three components of d to be complex According to Sect 4 1 3 only the vertical components of k and k can have imaginary parts However to account for a tilt of the particle it may be necessary to evaluate F q with a rotated scattering vector d that has complex q or du The following tables summarize the implemented particle geometries roughly ordered by decreasing symmetry Afterwards the detailed documentation is in alpha betical order Shape Name Symmetry Parameters Reference Q FullSphere Rz R Page 48 a FullSpheroid Dosh kR H Page 52 u Cylinder Dooi R H Page 44 30 GivoqaeacEevgqgeced TruncatedSphere TruncatedSpheroid Cone TruncatedCube Prism6
51. vo YAT H 50 Examples IF v Figure B 20 Normalized intensity F V computed with R 10 nm R 3 8 nm and H 3 2 nm for four different angles w of rotation around the z axis References Agrees with the IsGISAXS form factor Anisotropic hemi ellipsoid 7 Eq 2 42 with wrong sign in the z dependent phase factor or Hemi spheroid 8 Eq 229 ol B 10 FullSpheroid Real space geometry 2R Perspective Top view Side view Figure B 21 A full spheroid generated by rotating an ellipse around the vertical axis Syntax and parameters FormFactorFullSpheroid radius height with the parameters e radius R e height H Form factor etc Notation Az wa Te s dz dy Results fe J1 q R F 4rexp iq H 2 dz Rg _ 0 qf 2 V R H 2 k S 7R 02 Example 5 4 E e E J S 1 E Y 1 2 3 4 5 dl Figure B 22 Normalized intensity F V computed with R 3 5 nm and H 9 8 nm References Agrees with the Full spheroid form factor of IsGISAXS 7 Eq 2 37 8 Eq 227 with corrected volume formula We also discovered a wrong factor of 2 in the IsGISAXS code 59 B 11 Prism3 triangular Real space geometry Perspective Top view Side view Figure B 23 A prism based on an equilateral triangle Syntax and parameters FormFactorPrism3 length height with the parameters e length of one base e
52. xis References Agrees with Cuboctahedron form factor of IsGISAXS 7 Eq 2 34 8 Eq 218 except for different parametrization L 2R scrsaxs 43 B 6 Cylinder Real space geometry y 2R Z X H X Perspective Top view Side view Figure B 13 An upright circular cylinder Syntax and parameters FormFactorCylinder radius height with the parameters e radius of the circular base R e height H Form factor etc Notation 4 y2 a Results H H J R F rR H sine 4 7 exp Ln KS V 7R7H S 7R 44 Examples Ha V Figure B 14 Normalized intensity F V computed with R 3 nm and H 8 8 nm for four different tilt angles 9 rotation around the y axis References Agrees with Cylinder form factor of IsGISAXS 7 Eq 2 27 8 Eq 223 45 B 7 EllipsoidalCylinder Real space geometry Perspective Top view Side view Figure B 15 A upright cylinder whose cross section is an ellipse Syntax and parameters FormFactorEllipsoidalCylinder radius_a radius_b height with the parameters e radius_a in x direction R e radius_b in y direction R e height H Form factor etc Notation Y y Us Ra dye Results H H J F 2rR R H exp L sinc i V TR RA 46 Examples 5 LU Ha V o 90 10 107 107 10 104 10 10 107 10 10 10 0 3 4 5 9 iC 0 1 2 Figure B 16
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