A brief introduction to SAXS M.H.J. Koch 1
SAXS ∆λ/λ ~0 r>>t very small!! For a single electron: + 2 2 1 cos 2 θ 1 ( ) r 2 2 θ 0 = = I e ( ) r I I 0 0 0 2 2 2 r r classical electron radius For SAXS this factor =1 I 0 =I in exp(- µ t) = 2.82 10 -15 m 2 sin2 θ = 2 θ cos 2 θ = 1
Electrons in an electromagnetic field are accelerated and therefore emit radiation: they scatter. The spatial distribution of the scattered intensity depends on the geometry of the experiment. For unpolarized incident radiation the spatial distribution on the equator is: 2 4 1 cos ( 2 θ ) 1 ω + 2 I(2 ) r I (1) θ = ⋅ ⋅ ⋅ 0 0 2 2 2 2 2 r ( ω ) ω − 0 where 2 θ is the angle between the incident and the scattered beam. corresponds to the k ω = 0 Obs. m natural frequency 2 Ө ( ν 0 = ω 0 /2 π) of the oscillator and ω to the frequency of the incident radiation. 3
The most interesting factor in the previous equation is the one describing the frequency dependence. For the AMPLITUDE (E with I= E·E*) 2 ω 2 2 ( ω ω ) − 0 The natural frequency of the oscillator ( ω 0 ) corresponds to the binding strength of electrons in atoms and lies somewhere in the UV to X-ray region. If the incident radiation is visible light ( λ ≈ 500 nm), ω << ω 0 and the factor reduces to: 2 ω 2 ω 0 The amplitude of the scattered radiation at r is proportional to ω 2 and in phase with the incident radiation. This is Rayleigh scattering . The scattered intensity is proportional to ω 4 hence the blue sky. 4
X-Rays If the incident radiation is X-rays ( λ ≈ 0.1 nm), ω 0 << ω and the factor 2 ω = -1 2 2 ( ω ω ) − 0 The scattering amplitude is independent of the frequency and its phase is shifted by 180 degrees relative to the incident radiation. This is Thomson scattering As it is independent of the frequency of the incident radiation, the world of X-rays is colorless with shades of gray (i.e. contrast) only and Eq.1 above simplifies to: 2 1 cos ( 2 ) 1 + θ 2 I(2 ) r I θ = ⋅ ⋅ 0 0 2 2 r 2 e 15 r 2 . 817 10 classical electron radius − = = m 0 2 mc 5
Energy out /energy in 1 2 I (0 ) r I For unpolarized X-rays at 2 Ө = 0: = e 0 0 2 r Detector d σ I (2 θ ) 2 r 2 b e = ⋅ = I e (2 θ ) d � I r 2 d Ω 0 b : scattering length = r 0 = 2.810 -15 m for the electron sample I 0 The differential scattering cross-section r has the dimension of an area and represents 1m 2 Energy scattered/unit solid angle/unit time Energy incident/unit area/unit time For one electron: the amplitude of scattering |I e (0)| 1/2 = 2.810 -15 |I 0 | 1/2 and as the scattering amplitude ≡ f = amplitude scattered by an object f e =1 amplitude scattered by an electron in identical conditions 6
Particle size (D) and wavelength of the radiation λ >> D Light scattering λ < D X-ray scattering Observer Observer When λ >> D all N electrons in the particle are accelerated in phase the scattering amplitude is N times that of one electron. When λ < D the electrons in the particle are no longer moving in phase and one has to take the phase shift of the waves into account. 7
Waves and Interference Interferences lead to fringe patterns. This is illustrated here with water waves. When “solving” a structure the problem is to go from the fringe pattern – in the case of X-ray diffraction from the intensities of the fringes – to the distribution of sources i.e. of scatterers. Similar effects are observed with optical transforms obtained by shining coherent visible (laser) light through small apertures (see e.g. Cantor and Schimmel , Biophysical Chemistry, Part II, Ch. 13). 8
Interference and coherent scattering In Thomson (coherent scattering) the scattered wave is 180° out of phase with the incident wave. r . S S / λ r ( ) S − S 0 = s λ 2 Θ r . S 0 2sin θ = S 0 / λ s λ Path difference = ⋅ − ⋅ r S r S 0 2 π Phase difference = (r ⋅ S − r ⋅ S ) 0 λ The total amplitude from two centers (one at the origin and one at r ) is thus: 2 = ∑ F(s) f exp(2 π i sr ) f f exp(2 π i s r ) = + ⋅ e i e e 2 9 i 1 =
The sum of amplitudes for N electrons: N = ∑ F( ) f exp(2 π i ) s s ⋅ r e i i 1 = F( s ) is the Fourier transform of the distribution of electrons The average of the exponential factor over all orientations of r relative to s for randomly oriented particles (e.g. in solution) is: sin ( 2 ) π sr exp ( 2 i ) π < s ⋅ r > = i 2 π sr As this is a real number there is no phase problem but one has lost most of the structural information. 10
Scattering factor For an atom with a continuous For λ = 0.15 nm f’ = 0.3641 radial electron density ρ (r): f” = 1.2855 ∞ sin(2 sr ) π ∫ 2 F(s) f(s) 4 ρ (r) r dr π ≡ = 2 sr π 0 and since sin x f(0) = Z lim 1 = x 0 x → For modeling purposes one often uses larger spherical subunits (beads, dummy residues, etc) for which: 3 ( ( ) ( ) ) ( ) sin 2 2 cos 2 11 = π − π π f s sR sR sR 3 sphere ( ) 2 π sR
Anomalous scattering Near an absorption edge, the dissipative effects due to the rearrangement of the electrons can no longer be neglected. The scattering factor f e must be modified to take anomalous scattering into account f e = 1 + f e ’ + if e ” (f” is always π /2 ahead of the phase of the real part) ω A f " ( ω , 0) � ( ω ) µ : linear (photoelec tric) absorption coefficien t, = 4 π r cN 0 A N : Avogadro' s number, A atomic weight A 2 ω ' f " ( ω ' ) ∫ f ' ( ω ) d ω ' Kramers Kronig = − 2 2 ω ' π ω − Note that for most practical purposes, f’ and f” are independent of s = 2sin θ/λ 12
Selenium f‘ and f‘‘ ω A f " ( ω , 0) ( ) � ω = 4 π r cN 0 A 2 ω ' f " ( ω ' ) ∫ f ' ( ω ) d ω ' = 2 2 ω ' π ω − SeK Note the sign and absolute value of the corrections. 13
Scattering from N spherical atoms: N = ∑ F( ) f (s)exp(2 π i ) ⋅ s s r i i i 1 = F( s ) is the Fourier transform of the distribution of the spherical atoms. Crystallographers call this the structure factor. Note that in SAXS the structure factor refers to structure of the solution. The intensity is, of course: Crystal structure - N N ∑∑ I( ) f (s)f (s)exp(2 π i ( - ) ) = ⋅ s s r r atomic coordinates i j j i i 1 j 1 = = This is a real number! For random orientation sin ( 2 s r ) π ij and exp ( 2 π i ( - ) < s ⋅ r r > = j i 2 π s r ij Solution – sin(2 π sr ) N N I(s) ∑∑ ij f (s)f (s ) = Distance distribution only! i j 2 π sr 14 i 1 j 1 = = ij Debye (1915)
sin(2 π sr ) N N I(s) ∑∑ SAXS: ij f (s)f (s ) = i j 2 π sr i 1 j 1 ij = = In isotropic systems, each distance d = r ij contributes a sinx/x –like term to the intensity. A scattering pattern is a continuous function of s. Short distances >> low frequencies dominate at high angles Large distances >> high frequencies contribute only at low angles. 15
The wider a function in real space the narrower its transform in reciprocal space 1) The Fourier transform of the Dirac delta function ∞ ∫ δ (0) δ (x)dx 1 = ∞ = − ∞ is the 1(x) function (i.e. the function which has a constant value of 1 over the interval [- ∞ , ∞ ]. 2) Obviously, the Fourier transform of 1(x) is δ (x). 3) The Fourier transform of a Gaussian is also a Gaussian π 2 2 2 (exp( )) ( ) exp( / ) − = = − π FT ax F k k a a Note the relationship between the widths. If the Gaussian has a width σ R =(1/2a) ½ , its transform has a width σ F =(a/2 π 2 ) ½ and σ R σ F =1/2 π . The δ -function is an infinitely narrow Gaussian. 16
Kinetics of the Ca 2+ -dependent swelling transition of Tomato Bushy Stunt Virus 0.1 s ID2 (ESRF) Perez et al. 10.0 Intensity 1.0 0.1 0.02 0.04 Q (Å -1 ) 0.06 Larger objects scatter at lower angles! 17
In an ideal solution The solute particles are randomly oriented and their positions are uncorrelated in space and time. Consequently their scattering in isotropic and incoherent. The total scattering intensity is the sum of the coherent scattering intensity of all molecules. It is a function of the scattering angle or modulus of the scattering vector only: I(s). Usually one plots log(I(s)) vs s, because the intensity falls off rapidly due to the interferences. sin(2 π sr ) N N i.e. for atoms one can neglect the ∑∑ ij i (s) f f = 1 i j 2 π sr s-dependence of f i i 1 j 1 ij = = If one uses a continuous density distribution ρ (r) this becomes sin( 2 ) π sr ∫∫ ( ) ( ) ( ) 12 = ρ ρ i s r r d r d r 1 1 1 2 2 1 2 2 π sr V 12 18
Interactions of X-rays with matter fluorescence Incoherent scattering I 0 Transmitted beam Sample Incident beam I=I 0 exp(- µ t) Coherent scattering Structural information at the atomic/molecular level is in: coherent scattering and to a limited extent in absorption/fluorescence near edges (EXAFS, XANES) At lower resolution transmission/phase contrast imaging is also useful 19
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