Fourier Synthesis in Chemical Analysis

Transcription

1 Fourier Synthesis in Chemical Analysis Ivan Vicković Chemistry Dept, Faculty of Science, Univ. of Zagreb Horvatovac 102A CEEPUS Summer School 10th International Symposium and Summer School on Bioanalysis Zagreb, July 7-14,

2 Why Fourier synthesis? mathematical method extensively applied in both natural and technical sciences Condition: a phenomenon can be described by periodic function Progress In Electromagnetics Research Symposium 2007, Prague, Czech Republic, August Electrical Vibrations of Yeast Cell Membrane M. Cifra, J. Vaniš, O. Kučera, J. Hašek, I. Frýdlova, F. Jelínek,J. Šaroch, and J. Pokorný Department of Electromagnetic Field, Czech Technical University and other institutions A bridge over Hulan River near Tieli city in Heilongjiang province collapsed due to overweighted trucks vibrations 2

3 What Chemical analysis? 1/2 this lecture is limited to 1. Cristallography, i.e. X-ray structure analysis There is periodicity of crystal lattice in solid state: 3

4 What Chemical analysis? 2/2 2: FT spectroscopy (examples FTIR, FTNIR, FTNMR, EXAFS, MS.) Dispersive spectroscopic instruments Spectrum is measured in frequency domain vs.?periodicity? FT spectroscopic instrument Michelson interferometer Cosine wave periodically varies from max to min amplitude depending on movable mirror, what corresponds to a path-length difference Interferogram is measured in time domain (SPECTRUM TO BE CALCULATED) 4

5 Why FT for Fourier synthesis? FT stands for Fourier Transform meaning: Fourier synthesis can be calculated through Fourier series or more generally through Fourier integral The Fourier series is defined through the coefficients C n f ( x ) = n = C n e in ω x The Fourier integral is defined through the Fourier transform F(ω)! f A 1 ( x) = lim e A 2π A iωx F( ω) dω In practice the synthesis is calculated as the series of Discrete Fourier Transform, but generally accepted expresions are: Software based on FFT algorithm (Fast Fourier Tranform) Fourier transform spectroscopy and different FT instruments Fourier transformation is applied in molecular and crystal structure analysis 5

6 Fourier transformationf as applied in diffraction experiments F Periodic arrangement Measured data: diffraction intensities of diffracted waves Obtained after correction of measured data: Amplitudes of diffracted waves Unmeasurable data obtained SOMEHOW: Phases of diffracted waves Diffracted waves (STRUCTURE FACTORS): defined by amplitudes and phases FOURIER SYNTHESIS applied to all structure factors results in map of distribution of electron density and molecular and crystal structure determination Interpretation:Location of maximums in the map represent atomic coordinates 6

7 X-ray structure analysis Chemical synthesis C O S Hg Diffraction power f a Crystal planes hkl To observe Diffraction wave + Intensity Phase I + ϕ Wave function F To calculate F Electron density map ρ FOURIER TRANSFORMATION List of atoms Atomic scattering factors Incomplete picture in reciprocal space Complete picture in reciprocal space Impossible to record Structure fator Complete picture in REAL SPACE PHASE PROBLEM Patterson function Direct methods 7

8 Fourier transformation as applied in CW-IR spectroscopy (FT-IR) Spectrometer with movable mirror and interferometer N.B: Do not mix up 1.continuous wave (CW due to contiuous mirror moving) and 2. continuous source (both mirrors fixed, not used in chemical analysis) F A cosine wave that varies from maximum amplitude to minimum amplitude every λ/2 movement of the movable mirror (on one of two arms) which corresponds to a pathlength difference of λ/2 Interferogram in time domaine Spectrum in frequency domain obtained after FT, corresponds to that obtained by a continuum source 8

9 Fourier transformation as applied in pulse NMR spectroscopy Frequency content of the pulse Single burst of radiofrequency energy N.B.: Do not mix up 1.continuous wave in CW-NMR working with oscilating magnetic field, measured in frequency domain and 2. continuous wave in CW-IR (CW due to continuous mirror moving), measured in time domain The nuclei shocked into oscillation by means of radiofrequency pulse generator Free-induction decay or interference pattern in time domain F Spectrum in frequency domain 9

10 That was about where we use Fourier transformations in chemical But what is it? analysis. Why it can be applied in fundamentally different areas of chemistry and, more generally, science and technology. What are domains and/or spaces? What are the basis of the method? 10

11 Important Fouriers Jean Baptiste Joséf Fourier ( ): Laplace was his teacher, he succeeded Lagrange in École Polytechnique. mathematician, politician, governator of Egypt worked on periodic functions (today called harmonic analysis) for the purpose of solving the heat equation in a metal plate without precise notion of function and integral in the early nineteenth century François Marie Charles Fourier ( ): philosopher, utopian-socialist, inspired the founding of several communities in USA (Texas, New Jersey, Ohio, New York State), he originated the word feminism in 1837, In the mid-20th century, Fourier's influence began to rise again among writers reappraising socialist ideas outside the Marxist mainstream. 11

12 Short history of application of Fourier synthesis in chemical analysis 1912/13 Braggs anticipated the application of Fourier synthesis in electron density calculation and crystal and molecular structure solving 1924 Epstein i Ehrenfest - application of Fourier synthesis in X-ray analysis 1934 Patterson phase problem solved as autoconvolution of electron density about 1970 application of Fourier Synthesis in spectroscopy : FTIR, FTNMR, FTICR about1980 pattern recognition, NMR imiging(2d,3d), MS etc

13 Fourier series Observe a periodic function with period p: f ( x) = e inωx = e inω( x+ p) The next sum is periodic too, having the same periode p: f ( x ) = n = C n e in ω x This series is called Fourier series if it satisfies the Fourier theorem 13

14 Fourier theorem If a function is square-integrable on the interval [ π,π], then the Fourier series converges to the function at almost every point. If f(x) is real-valued of real arguments, periodic with the period p if it is posible to find coefficients C n by means of the integral C n = 1 p c+ p f ( x ) e c -inωx dx 14

15 Fourier integral 1/3 To find out what is the Fourier integral let have the Fourier series in a limited interval: f x C e in ω x ( ) = for -T<x<T n n = - where C n is already known as: C n = 1 2T T f( t) e -inω t -T dt In order to analyse nonperiodic function let T 15

16 Fourier integral 2/3 For practical reasons observe the limes of finite sum: f k inωx lim Cne lim k k n= -k ( x) = = S k Process the finite sum: insert the substitution AL k =, un = π nω introduce C n into S k with infinitesimaly small values (angular frequences nω=nπ/t are very close with a distance of ω = πt the sum will transit to integral 0 16

17 Fourier integral 17 = 2 1 = ω π ω ω dtd t f e e x f t i x i A A A ) ( lim ) ( - = 2 1 = ω ω π ω d e x F i A A A ) ( lim dt t f e F t i ) ( ) ( ω ω - = Fourier transform defined as a function under Fourier integral: 3/3

18 function shown in terms by means of for set of frequences Fourier SERIES f sumation over n discret COMPARISON Fourier INTEGRAL in x x ω 1 iωx ( ) = C n e f( x) = e F( ω) dω n = - 2π - in x e ω e iω x integration overω continuous over COEFFICIENTS C n TRANSFORM F(ω) 18

19 Fourier transform and inverse Fourier transform It the Fourier transform F(ω) of the function f(x) is known: F ( ω) = ( t) dt -1 1 f( x) = F 2π e -iωt then this function f(x) can be obtained by means of inverse Fourier transform: f { } iωx F( ω) = e F( ω) dω F(ω) and f(x) are pair of Fourier transforms representing the same phenomenon in two different domains. 19

20 Symbolism f(x) F(u) f(x) = F {F(ω)} inverse Fourier transforms pair of Fourier transforms F(ω) =F {f(x)} Function which satisfies generalised Fourier theorem can be presented in two equivalent ways: in two different domains (time vs. frequency) in two different spaces (real vs. reciprocal, real vs. modul) (as a matter of fact, in two different coordinate systems in the real space!) Relation between a function and its transform shows the way how the domains are corresponding 20

21 Convolution (Faltung) The operation of convolution: c ( u) = f( u) g( u) = f ( r) g( u - r) dr Both functions f and g have the same argument u, but they are observed in different points r and (u-r) with distance u between them Autoconvolution is for : g(u)=f(u) Example:Patterson function which is autoconvolution of electron density, helps to solve phase problem) P ( xyz) = ρ ( r) ρ( u - r) dv N.B. The electron density itself is one half of pairs of Fourier transforms and the other half is: structure factor F(hkl) 21

22 Visualization of convolution of two boxcar functions g(x) moving over f(x) makes overlap f(x) g(x) Move g(x) to overlap f(x) Convolution is an integral that describes the amount of overlapping of two functions while one of the functions is moving over the other. 22

23 Visualization of convolution in diffraction experiments CRYSTAL = MOTIF LATTICE Convolution of a duck and a lattice corresponds to a molecule convoluted to crystal lattice 23

24 Convolution theorem 1/2 The operation of convolution is defined as: c ( u) = f( u) g( u) = f ( r) g( u - r) dr What if these functions have Fourier transform? 2πisr ( u r F { f r) } = F( s) = f( r) e dr F { g( u-r) } = G( s) = g( u-r) e 2πis - ) dr ( Then, their convolution c(u) has its Fourier transform C(s): F { } 2πisu c u) = C( s) = c( u) e du ( - F { } -2πisu C s) = c( u) = C( s) e ds ( - 24

25 Convolution theorem 2/2 If the function c(u) is the convolution of two functions f (u) and g(u) which have their Fourier transforms F(s) and G(s) c ( u) = f ( u) g( u) = f ( r) g( u - r) dr N.B. Here, the sign is used to notify the operation of convolution. There is no internatinaly recognized sign. Then the function c(u) has Fourier transform C(s) which is the product of Fourier transforms F(s) and G(s) C ( s) = F( s) G( s) Very useful attribute in data processing: difficult integral such as convolution can be substituted by simple linear multiplication of Fourier transforms! 25

26 Excited systems An interaction between a sample and electromagnetic field can give an information about the structure of the sample. The sample can be treated as a system excited by an input signal and the interaction can be described by an operator that reproduces input signal to the output one x(t) Input signal Limitations: Φ Excited system Φ Linear time-invariant operator y(t) Output signal System is linear and y(t) is linear combination: y(t)=φ{x(t)} Electromagnetic interaction is a periodic function and Fourier theorem may be applied and the output signal treated in two different domains! y(t)= Φ{x 1 (t)+ x 2 (t)}=φ{x 1 (t)} + Φ{x 2 (t)}= y 1 (t)+y 2 (t) There is no difference if input signals are simultaneous or separated in time. 26

27 General approach to the excited systems: Linear time-invariant system theory It investigates the relationship between input and output signal, i.e. frequency response of a linear and time-invariant system to an arbitrary input signal. It comes from applied mathematics and it is applied in communication theory, signal procesing, control theory, i.e. in electrical engeenering, image processing, social sciences, economics, Statistics, Etc. Image processing, holography scientifique measurments, electrochemistry, diffusion, diffraction, spectroscopy IR & NMR cyclotron resonance, nuclear quadrupole resonance, electron spin resonance, dielectric response, microwave response, muon spin rotation, 27

28 Frequency response: Relationship between the time domain and the frequency domain x(t) input signal X(s) input signal transform y(t) output signal Y(s) output signal transform h(t) - operator (impulse response) that reproduces input signal to the output one in time domain H(s) operator (transfer function) that reproduces input signal to the output one in frequency domain Operators h(t) and H(s) are related by Laplace transformation. They represent the same phenomenon in two different domains. It is often easier to do 1) the transforms, 2) multiplication, and 3) inverse transform than the original convolution!! The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = iω or s = 2πfi 28

29 Harmonic function as input signal Structural investigations are limiting case in the linear time-invariant systems since an input signal is the harmonic function originated in an electromagnetic field: iω t x ( t ) = e The output signal is a convolution of the impuls response function and the input signal harmonic function + i ω( t -τ ) y( t) = Φ { x( t) } = h( t) x(t) = h( t) e dt or, after Fourier transforming the impuls response h(t) from the time domaine to the transfer function H(ω) in the frequency domain: + H( ω ) h( t) e = - the output signal can be presented as a linear multiplication of the transfer function and the input signal: y( t) = H(ω) e iωt dt iωt - dt 29

30 Dirac s function Very usefull functions in data processing: Dirac s delta function is usually drawn as a vertical line, it is convenient for narrow pulse which has a height of 1 only at the exact point when t = 0 + for x δ ( x) = { 0for x 0 Dirac s comb is a series of unique pulses with period T. It has sampling property. Dirac s comb is Fourier transform to itself, up to the multiplicative constant. = 0 delta(t) time, t 30

31 Fourier transform domains in molecular structure analysis Linear timeinvariant system Harmonic function FT spectroscopy Time domaine Impuls response Time domaine Frequency domaine Transfer function CW-IR Interferogram Complex spectrum Pulse NMR FID (free induction decay) Frequency domaine Complex Spectrum Diffraction Reciprocal space Real space X-ray 1 i ω e ω t 1 iωt h( t) = H( ) dω 2π H( ω) = h ( t) e dt 2π F(hkl) Structure factor Electron density Electron Idem Electrostatic potential Neutron idem Nuclear potential 31

32 The use of X-Rays and Fourier transformations for molecular and crystal structure determination ( ) X-rays can be scattered and diffracted by crystals and visualized in terms of reflections from planes of atoms Intensities of x-rays reflected from different planes divers due to number of atoms and number of electrons in the atom lying in the reflecting plane William Lawrence Bragg William Henry Brag the use of X-rays as an instrument for the systematic revelation of the way in which crystals are built was entirely due to the Braggs Nobel prise

33 ( ) ( ) N S = C I S πi( hx + ky lz )) F exp(2 o j j + F ϕ( S) = 2πSr c HOW TO PRESENT THE PHENOMENON OF DIFFRACTION? * I ( S) = F( S) F( S) = Structure factor (in reciprocal space) r ( xyz) ( hkl) j= 1 ( ) N ( ) S = f ( S ) exp 2πiS r * j =1 * * j = xj a + y jb + S = ha + kb + lc * j z * j c * F 2 Number of diffraction intensities observed, phases LOST j j observed calculated Radiusvector in real space Radiusvector in reciprocal space F ( S) = f ( I, ϕ) F( S) = ρ ( r) e r 2πiSr 33 dv

34 Bragg s approach to the diffraction condition A d C d distance between the lattice planes in the real space B Path difference: AB+BC=δ AB=δ/2 sinθ=ab/d Reflection on the planes Bragg s law: sinθ=δ/2d 2dsinθ=nλ 34

35 Convolution in diffraction I( * S ) = F( S) F( S) = experiments F( S) 2 (Contribution to phase problem solving) Substitution: I( S ) = u = r r r ρ ( r) e 2πi S r 2πiSu I( S) = ρ( r) ρ( u+ r) e dvdv= u I( r S ) _ P( r) dv r ' ρ ( r' ) e P ( u) e u _ -2πi S r ' 2πiSu dv dv 1 st pair of Fourier transforms represents the same phenomenon (distribution of interatomic distances) in two different spaces 35

36 Patterson function The relation was found: _ 2πiSu I( S) = ρ( r) ρ( u+ r) e dvdv= P ( u) e u and by definition P( u) = ρ ( r) ρ ( u + r) dv r r u 2πiSu dv _ Patterson function: an autoconvoluition of electron density function found at different points (at u distance) in the same coordinate system (real space) But in practice, since the electron density is 2 2πi ( hx+ ky+ lz) unknown, it is calculated as: P( xyz) = F( hkl) e In protein crystalography, the intensity F(hkl) 2 of diffracted maximums in Patterson function are combined with different additional information hkl 36

37 The meaning of Patterson function Mathematical: autocorellation of electron density function. Patterson function and diffraction intensities make a pair of Fourier transforms Physical: simultaneous electron density in two detached points in real space Crystallographic: interatomic vectors that have to be combined in a way to show the molecular structure what indirectly solve the phase problem y u r r r electron density map ρ(x) Real space Patterson map P(x) y u x x 37

38 The principles of structure compleating and refinement Structure factor in reciprocal space F( S) = V ρ( r) e 2πiSr dv Electron density in real (crystal) space 2 * ( ) πisr r F( S) e dv ρ = V * _ ρ( r) F( S) 2 nd pair of Fourier transforms represents the same phenomenon (distribution of electrons) in two different spaces. 38

39 Fourier transformation in diffraction In practice, the maps are calculated as discrete Fourier series, i.e. as FFT (Fast Fourier Transform) P( xyz) ρ( xyz) = = 2πi( hx+ ky+ lz) I( hkl) e hkl _ 2πi( hx+ ky+ lz) F( hkl) e hkl Connected by convolution: Connected by multiplication: experiments I( P( r) I( S) ρ( r) F( S) P( u) = ρ ( r) ρ ( u + r) dv * S ) = F( S) F( S) = r F( S) 2 39

40 The full use of Reciprocal space in diffraction structure analysis Data collection: To determine the unit cell in the real space and to precalculate all possible directions in reciprocal space in which the diffraction beams can be expected a, b, c, α, β, γ, set of hkl Structure solving, completing and refining: 1. Patterson map as FT of intensities, 2. electron density map as FT of structure factors, 3. data processing P( r) I( S) ρ( r) F( S) 40

41 Fourier transformation in IR spectroscopic experiments Albert A. Michelson ( ), Nobel prise 1907 Multiplex signal decoding Multiplex cos waves by non-dispersive device, i.e. interferometer IR interferogram in time domain F Spectrum in frequency domain 41

42 Fourier transformation in NMR spectroscopic experiments Single burst of radiofrequency energy Richard R. Ernst (1933), Nobel prise1991 Frequency content of the pulse The nuclei shocked into oscillation by means of radiofrequency pulse generator FT Free-induction decay or interference pattern in time domain FT can be used as: Pulse NMR signal decoding Different kinds of signal corrections F Spectrum in frequency domain 42

43 FT Spectroscopic signal decoding FT-IR (CW-IR) FT-NMR (Pulsed NMR) + 2πi xυ -2 x I ( x) = S ( υ ) e dυ S ( υ ) = I ( x) e πi υ dx - Intensity of interferogram Time domain x +x t Response pulse (interferogram) depends on path difference in two arms carrying fixed and movable mirror Response pulse (FID) decreases in time F F Spectrum Frequency domaine + - SPECTRUM SPECTRUM 43

44 Convolution in spectroscopic experiments Diffraction: convolution has the key role in structure solving: Patterson function Spectroscopy: convolution has the key role in data processing: 1. Spectrum correction Apodization Smoothing Trimming Sensitivity Resolution Truncation Amertization Contrast 2. Interferogram correction Phase correction Signal-to-noise ratio 3. Instrument error/deficiency corrections Zero filling Etc. 44

45 Example1. Spectrum filtering by weighting function convolution of spectrum and weighting function transform can be replaiced by a product of interferogram and weighting function spectrum F weighting function Corrected spectrum CONVOLUTION interferogram F = (spectrum) S ( υ) W( υ) = Scorr( υ) weighting function I( x) w( x) = I ( x) corr F Corrected intreferogram F = (Corrected spectrum) MULTIPLICATION 45

46 Example 2. Truncation A. Infinitely long cos wave B. Truncation function C. Truncated cos wave (multiplied A and B) D. FT of truncated wave has to be transformed to spectrum Example 3. Apodization A. Infinitely long cos wave B. Apodization function C. Apodized cos wave (multiplied A and B) (πόδι=foot, leg) D. FT of apodized wave has to be transformed 46

47 Example 3. Apodization (cont) x 1 Apodization { 1 - for - X function: a ( x ) = X 0 for - X > x -X +X x Fourier transform of apodizing function: x > + X + X F { } 2 sin ( πυ a ( x ) = A ( υ ) = X 2 ( πυ X X ) ) Filtered spectrum: But using Fourier transforms: and filtered spectrum can be obtained by simple Fourier transformation: + S f ( υ ) = S ( υ ) A( υ ) = S ( υ ) A( υ '-υ ) dυ I f ( x) = I ( x) a( x) it is easier to calculate S f ( υ ) = + - I f - ( x) e -2πiυ x dx 47

48 Example 4. Phase correction Sources of phase error: error in triggering start of experiment in t 0 etc. Possibble phase correction function: w(x) Its Fourier transform: W (υ ) Registered interferogram: can be corrected by multiplication: I(x) I corr See Example 1! ( x) = I( x) w( x) Registered spectrum is difficult to correct using convolution: S corr ( υ ) = S ( υ ) W ( υ ) But the corrected spectrum is easier to calculate from corrected interferogram by Fourier transformation: S corr ( υ ) = + - I corr ( x) e -2πiυ x dx 48

49 Résumé Application of Fourier transformation: Diffraction methods: scattering field density map (x-ray/electron density) Spectroscopy methods: multiplex signal decoding (interferogram/spectrum) Application of convolution: Diffraction methods: indirect determination of atomic positions in crystal (Patterson map) Spectroscopy methods: interferogram and spectrum corrections 49

50 Conclusion Linear respons theory explains why Fourier frequency analysis is applicable to systems excited by broadband electromagnetic radiation. Interaction between excited system and exciting field may be presented in two ways so called domains (time/frequency,coordinate/momentum), spaces (real/reciprocal, real/momentum), or coordinate systems In one domaine it is convenient to observe the interaction and measure the output signal, The other domain is suitable for understanding the output signal Two domains are communicating by means of Fourier frequency analysis 50