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Scattering of X-rays

Scattering of X-rays. Books on SAS. - " The origins" (no recent edition) : Small Angle Scattering of X-rays A. Guinier and A. Fournet, (1955), in English, ed. Wiley, NY - Small-Angle X-ray Scattering O. Glatter and O. Kratky (1982), Academic Press.

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Scattering of X-rays

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  1. Scattering of X-rays EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  2. Books on SAS • - " The origins" (no recent edition) : Small Angle Scattering of X-rays • A. Guinier and A. Fournet, (1955), in English, ed. Wiley, NY • - Small-Angle X-ray Scattering • O. Glatter and O. Kratky (1982), Academic Press. • - Structure Analysis by Small Angle X-ray and Neutron Scattering • L.A. Feigin and D.I. Svergun (1987), Plenum Press. • - Neutron, X-ray and Light scattering: Introduction to an investigative Tool for Colloidal and Polymeric Systems (1991), • P. Lindner and Th. Zemb (ed.), North-Holland • The Proceedings of the SAS Conferences held every three years are usually published in the Journal of Applied Crystallography. • The latest proceedings are in the J. Appl. Cryst., 30, (1997), next one soon to come out EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  3. Reminder - notations Fourier Transform F. T. r(r) F. T.-1 F(s) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  4. F. T. r(r + R) Properties of the Fourier Transform • 1 – linearity • FT (l1r1+ l2r2) = l1FT(r1) + l2 FT(r2) • 2 – value at the origin • 3 – translation EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  5. d(r) o r f(x) 1 -a/2 a/2 Fourier Transforms of simple functions • 1 – delta function at the origin • r(r)=d(r) • 2 – Gaussian of width 1/K • 1 – constant unitary value • F(s)=1 • 2 – Gaussian of width K 1 s • 3 – rectangular function • 3 – sinc a EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  6. Convolution product 1 B(r) A(r)*B(r) A(r) rA r rB r EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  7. Convolution product rA -rB B(r-u) A(r)*B(r) A(u) rA u rB u rA +rB EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  8. Fundamental Property of the convolution product EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  9. g0(r) r r Autocorrelation function spherical average r(r)=r (uniform density) => g(0)= r2V characteristic function particle  ghost 1 EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  10. rij j p(r) i Dmax r Distance distribution function g0(r) : probability of finding a point at r from a given point number of el. vol. i V - number of el. vol. j  4pr2 number of pairs (i,j) separated by the distance r  4pr2Vg0(r)=(4p/r2)p(r) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  11. Scattering by a free electron The elastically scattered intensity is given by the Thomson formula 2q : scattering angle,cos2q close to 1 at small-angles X-ray scattering length of an electron is the scattering cross-section of the electron I0 intensity (energy/unit area /s) of the incident beam. The scattered beam has an intensity Ie/d2 at a distance d from the scattering electron. In what follows, is omitted and only the number of electrons is considered EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  12. Scattering by an electron at a position r detector r.s0 s1 s0 s1 r source 2q s0 r.s1 O Path difference = r.s1-r.s0 = r.(s1 - s0) or r.(s1 - s0)/l in cycles, for X-rays of wavelength l EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  13. the scattering vector s scattered s1 s length 1/l s0 2q s is called the scattering vector O length 1/l The scattered amplitude by the electron at r is where E(s)is thescattered amplitude by an electron at the origin EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  14. ! scattering vector SAXS measurements are restricted to small-angles : 2q 5 deg = 85 mrad at l = 1.5 Å In this case, the further approximation is valid : EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  15. Scattered amplitude F(s) is the Fourier transform from the electron density r(r) describing the scattering object Scattered intensity Remark For a particle I(0) = m2 m : number of electrons EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  16. EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  17. Particles in solution The relevant quantity is now the contrast of electron density between the particle and the solvent Dr(r) = r(r) - r0 with r0 = 0.3346 el. A-3 the electron density of water EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  18. Contrast of electron density el. A-3 particle solvent EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  19. X-ray scattering power of a protein solution Let W be the probability of an incident photon being scattered by a solution of spherical proteins : using the expression for the optimal thickness d(cm) = 0.3/l2 (A3) Protein : solvent Example : 10 mg/ml solution of myoglobin v= 0.74 cm3.g-1 r=20A, l = 1.5A W=2 10-5 (x efficiency x geometrical factor) from H.B. Stuhrmann Synchrotron Radiation Research H. Winick, S. Doniach Eds. (1980) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  20. Solution of particles = * * Solution Dr(r) F(c,s) Motif (protein) Drp(r) F(0,s) Lattice d(r) d(c,s) * = . EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  21. Solution of particles For spherically symmetrical particles I(c,s) = I(0,s) x S(c,s) structure factor of thesolution form factor of theparticle Still valid forglobular particles though over a restricted s-range EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  22. Solution of particles - 1 – monodispersity: identical particles - 2 – size and shape polydispersity - 3 – ideality : no intermolecular interactions • 4 – non ideality : existence of interactions • between particles In the following, we make the double assumption 1 and 3 2 and 4 dealt with at a later stage in the course EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  23. Ideal and monodisperse solution Particles in solution => thermal motion => particles have a random orientation / X-ray beam. The sample is isotropic.Therefore, only the spherical average of the scattered intensity is experimentally accessible. Ideality and monodispersity EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  24. Let us use the properties of the Fourier transform and of the convolution product with EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  25. Or one of the symmetrical expressions : EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  26. Relationships real space – reciprocal space < > * r(r) p(r) p(r) FT FT FT . < > F(s) I(s) I(s) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  27. Debye formula If the particle is described as a discrete sum of elementary scatterers,(e.g. atoms) the scattered intensity is : where the fi(s) are the atomic scattering factors, and the spherically averaged intensity is (Debye) : where The Debye formula is widely used for model calculations EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  28. Debye formula If the particle is described using a density distribution, the Debye formula is written : EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  29. Porod invariant For r=0 : By definition : Q is called the Porod invariant Q depends on the mean square electron density contrast EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  30. ideal monodisperse Intensity at the origin is the concentration (w/v), e.g. in mg.ml-1 EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  31. ideal monodisperse Intensity at the origin If : the concentration c (w/v), the partial specific volume , the intensity on an absolute scale, i.e. the number of incident photons are known, Then, the molecular weight of the particle can be determined from the value of the intensity at the origin EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  32. ideal monodisperse Intensity on an absolute scale The experimental intensity Iexp(0) is expressed as : d : thickness of the sample a : sample-detector distance scattering cross-section of the electron I0 total energy of the incident beam EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  33. Intensity on an absolute scale The determination of I0 is performed using: - attenuation of the direct beam by calibrated filters (valid in the case of monochromatic radiation, beware of harmonics) - by using a well-known reference sample like the Lupolen (Graz) - using the scattering by water as proposed by O. Glatter in D. Orthaber, A. Bergmann & O. Glatter, J. Appl. Cryst. (2000), 33, 218-225 Thorough presentation of calculations on an absolute scale P. Bösecke & O. Diat J. Appl. Cryst. (1997), 30, 867-871 Clear presentation of the geometrical calculations and all the procedures used on ID2 (ESRF) to put intensities on an absolute scale. EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  34. ideal monodisperse Guinier law For small values of x, sinx/x can be expressed as : Hence, close to the origin: The scattering curve of a particle can be approximated by a Gaussian curve in the vicinity of the origin EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  35. ideal monodisperse Guinier law Let us introduce the expansion of sinx/x into the Debye formula : We obtain the following expression : EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  36. ideal monodisperse Guinier law where r0 is the center of mass Radius of gyration Guinier law EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  37. ideal monodisperse Radius of gyration Radius of gyration : Rg is the quadratic mean of distances to the center of mass weighted by the contrast of electron density. Rg is an index of non sphericity. For a given volume the smallest Rg is that of a sphere : Ellipsoïd of revolution (a, b) Cylinder (D, H) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  38. ideal monodisperse Guinier plot example The law is generally used under its log form : A linear regression yields two parameters : I(0) (y-intercept) Rg from the slope Validity range : 0 < 2pRgs<1.2 For a sphere EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  39. 4 5 10 30C 25C 4 4 10 20C 15C g-crystallins c=160 mg/ml in 50mM Phosphate pH 7.0 4 3 10 10C 4 2 10 4 1 10 0 0 0.01 0.02 -1 q)/l s = 2(sin A Attractive Interactions I(s) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001 A. Tardieu et al., LMCP (Paris)

  40. 4000 3500 92,5mg/ml 31,5 mg/ml 13,3 mg/ml 3000 6,6 mg/ml 3,3 mg/ml 2500 I(s) 2000 1500 1000 500 0 0 0.005 0.01 0.015 0.02 -1 q)/l A s = 2 (sin Repulsive Interactions ATCase in 10mM borate buffer pH 8.3 EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  41. Virial coefficient In the case of moderate interactions, the intensity at the origin varies with concentration according to : Where A2 is the second viral coefficient which represents pair interactions and I(0)ideal is  to c. A2 is evaluated by performing experiments at various concentrations c. A2 is  to the slope of c/I(0,c) vs c. EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  42. ideal monodisperse Rods and platelets In the case of very elongated particles, the radius of gyration of the cross-section can be derived using a similar representation, plotting this time sI(s) vs s2 Finally, in the case of a platelet, a thickness parameter is derived from a plot of s2I(s) vs s2 : T : thickness with EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  43. ideal monodisperse Volume of the particle Hypothesis : the particle has a uniform density and (Porod invariant) uniform density => Hence the expression of the volume for a particle of uniform density : EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  44. ideal monodisperse The asymptotic regime : Porod law Hypothesis : the particule has a sharp interface with the solvent with a uniform electron density Porod showed that the asymptotic behaviour of the scattering intensity is given by : S is the area of the solute / solvent interface B is a correction term accounting for : - short distance density fluctuations -uncertainties of i(s) at large s (weak signal) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  45. ideal monodisperse Distance distribution function • In theory, the calculation of p(r) from I(s) is simple. • Problem : I(s) - is only known over [smin, smax] : truncation • - is affected by experimental errors • - might be affected by distorsions due to the beam-size • and the bandwidth Dl/l (neutrons) • Calculation of the Fourier transform of incomplete and noisy data, which is an ill-posed problem. Solution : Indirect Fourier Transform. See lectures by O. Glatter and D. Svergun EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  46. Calculation of p(r) p(r) is calculated from i(s) using the indirect Fourier Transform method Basic hypothesis : The particle has a finite size p(r) is parameterized on [0, DMax] by a linear combination of orthogonal functions EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  47. ideal monodisperse Distance distribution function The radius of gyration and the intensity at the origin can be derived from p(r) using the following expressions : and This alternative estimate of Rg makes use of the whole scattering curve, and is much less sensitive to interactions or to the presence of a small fraction of oligomers. Comparison of both estimates : useful cross-check EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  48. ideal monodisperse p(r) :example Aspartate transcarbamylase from E.coli (ATCase) Heterododecamer (c3) 2(r2)3 quasi D3 symmetry Molecular weight : 306 kDa Allosteric enzyme 2 conformations : T : inactive, compact R : active, expanded T Rcrys Rsol EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  49. ideal monodisperse ATCase : scattering patterns EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

  50. ideal monodisperse ATCase : p(r) curves DMax EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

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