Fourier synthesis in chemical analysis CEEPUS Zagreb-2010€¦ · Fourier Synthesis in Chemical...

FourierSynthesisin

Chemical Analysis

CEEPUS Summer School in Zagreb July 13, 2010

1

Ivan VickovićChemistry Dept, Faculty of Science, Univ. of Zagreb

Horvatovac [email protected]

CEEPUS Summer School10th International Symposium and Summer School on Bioanalysis

Zagreb, July 7-14, 2010

• mathematical method extensively applied in both natural and technical sciences

• Condition: a phenomenon can be described by periodic function


2

Why Fourier synthesis?

A bridge over Hulan River near Tieli city in Heilongjiang province collapsed due to overweighted trucks vibrations

Progress In Electromagnetics Research Symposium 2007, Prague, Czech Republic, August 27-30 215

Electrical Vibrations of Yeast Cell Membrane

M. Cifra, J. Vaniš, O. Kučera, J. Hašek, I. Frýdlova, F. Jelínek,J. ·Šaroch, and J. PokornýDepartment of Electromagnetic Field, Czech Technical University and other institutions

this lecture is limited to

1. Cristallography,

i.e.

X-ray structure analysis


3

There is periodicity of crystal lattice in solid state:

What Chemical analysis? 1/2

2: FT spectroscopy

(examples FTIR, FTNIR, FTNMR, EXAFS, MS...)


4

Dispersive spectroscopic instruments

FT spectroscopic instrument –Michelson interferometer

vs.

Spectrum is measured in frequency domain

Interferogram is measured in time domain (SPECTRUM TO BE CALCULATED)

?PERIODICITY?

Cosine wave periodically varies from max to min amplitude depending on movable mirror, what corresponds to a path-length difference

What Chemical analysis? 2/2

Why FT for Fourier synthesis?


5

The Fourier integral is defined through the Fourier transform F(ω) !

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

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 Cn∑

∞

∞=

=n

xinn eCxf ω)(

ωωπ

ω dFexf xi

A

AA

)(lim)(∫

∞→ 21

=

Fourier transformation as applied in diffraction experiments


6

Measured data: diffraction intensities of diffracted wavesObtained after correction of measured data: Amplitudes of diffracted wavesUnmeasurable 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

FPeriodic arrangement

F

⇒++

Complete picture in REAL SPACE

C

O

S

Hg

....

....

....

List of atoms

Chemical synthesis

+

hkl..................

Crystal planes

⇒

fa

...

...

...

...

...

...

Atomic scattering factors

Diffraction power

Incomplete picture in reciprocal space

ρ

⇔

Electron density map

FOURIERTRANSFORMATION

PHASE PROBLEM

Patterson function Direct methods

F

..

..

..

..

..

..

Wave function

Structure fator

To calculate

Impossibleto record

Complete picture in reciprocal space

To observe

I

..

..

..

..

..

..

ϕ

..

..

..

..

..

..

Intensity Phase

Diffraction wave

7

X-ray structure

analysis


F


8

Fourier transformation as applied in CW-IR spectroscopy (FT-IR)

Spectrometer with movable mirror and interferometer

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 path-length difference of λ/2

Interferogram in time domaine

Spectrum in frequency domain obtained after FT, corresponds to that obtained by a continuum source

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


9

Spectrum in frequency domain

Free-induction decay or interference pattern in time domain

The nuclei shocked into oscillation by means of radiofrequency pulse generator

Single burst of radiofrequency energy

Frequency content of the pulse

F

Fourier transformation as applied in pulseNMR spectroscopy

N.B.: Do not mix up 1.continuous wave in CW-NMR working with oscilating magnetic field, measured in frequency domainand2. continuous wave in CW-IR (CW due to continuous mirror moving), measured in time domain


10

�But what is it?

�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?

That was about where we use Fourier transformations in chemical

analysis.


11

Jean Baptiste Joséf Fourier (1768-1830): �Laplace was his teacher, �he succeeded Lagrange in École Polytechnique.�mathematician, �politician, �governator of Egypt 1798-1800�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 (1772-1837):� 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.

Important Fouriers

Short history of application of Fourier synthesis in chemical analysis


12

•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.------------

Fourier series


13

Observe a periodic function with period p:

The next sum is periodic too, having the same periode p:

)()( pxinxin eexf +== ωω

∑

∞

∞=

=n

xinn eCxf ω)(

This series is called Fourier series if it satisfies the Fourier theorem

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 Cn by means of the integral


14

dxexfp

C xin

pc

cn

ω-∫

)(

+1

=

Fourier integral


15

To find out what is the Fourier integral let have the Fourier series in a limited interval:

xin

nn eCxf ω

∑

∞

∞-=

=)( for -T<x<T

where Cn is already known as: dtetfT

C t

T

Tn

ωn∫

i-

-

)(21

=

1/3

In order to analyse nonperiodic function let

∞→T

Fourier integral For practical reasons observe the limes of finite sum:


16

kk

xin

k

knn

kSeCxf

∞→∞→

∑ limlim)( ===

ω

-

Process the finite sum:

� insert the substitution

� introduce Cn into Sk� with infinitesimaly small values (angular frequences nω=nπ/T are very close with a distance of�the sum will transit to integral

ωπ

nuAL

k n == ,

2/3

0→= Tπω∆

Fourier integral


17

=21

= ωπ

ωω dtdtfeexf tixi

A

AA

)(lim)( -

∞

∞

∞→

∫∫

=21

= ωωπ

ω dFe xi

A

AA

)(lim∫

∞→

dttfeF ti )()( ωω -

∞

∞

∫=

Fourier transform defined as a function under Fourier integral:

3/3


18

Fourier SERIES Fourier INTEGRAL

function

shown in terms

by means of

sumation overn integration overω

for set of frequences

discret continuous

over COEFFICIENTS Cn TRANSFORM F(ω)

xin

nn eCxf ω

∑

∞

∞-=

=)( ωωπ

ω dFexf xi )()(∫

∞

∞

21

=-

xine ω xie ω

COMPARISON


19

Fourier transform and

inverse Fourier transform

{ } ωωπ

ω ω dFeFxf xi )()()(∫

∞

∞

21

== 1-F

dttfeF ti )()( ωω -

∞

∞

∫=

It the Fourier transform F(ω) of the function f(x) is known:

then this function f(x) can be obtained by means of inverse Fourier transform:

F(ω) and f(x) are pair of Fourier transforms representing the same phenomenon in two different domains.

Symbolism


20

f(x) F(u) inverse Fourier transformsf(x) = F { F(ω)}

F(ω) =F { f(x)}

pair of Fourier transforms

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

Convolution (Faltung)


21

The operation of convolution:

drrugrfugufuc )()()()()( -¤∫

∞

∞

•==

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)

dVrurxyzP )-()()(∫

∞

∞

ρρ •=N.B. The electron density itself is one half of pairs of Fourier transforms and the other half is: structure factor F(hkl)


22

g(x) moving over f(x) makes overlap f(x)¤g(x)

Move g(x) to overlap f(x)

Visualization of convolution of two boxcar functions

Convolution is an integral that describes the amount of overlapping of two functions while one of the functions is moving over the other.

Visualization of convolutionin diffraction experiments


23

Convolution of a duck and a lattice corresponds to a molecule convoluted to crystal lattice

CRYSTAL = MOTIF ¤ LATTICE

Convolution theorem


24

The operation of convolution is defined as:

drrugrfugufuc )()()()()( -¤∫

∞

∞

•==

What if these functions have Fourier transform?

{ } dsesCucsC isuπ2== -)()()(∫

∞

∞-

F

{ } drerfsFrf isrπ2== )()()(∫

∞

∞-

F { } ( )drerugsGrug ruis -

∞

∞

∫

π2

+

== )()()( ---

F

Then, their convolution c(u) has its Fourier transform C(s):

{ } dueucsCuc isuπ2== )()()(∫

∞

∞-

F

1/2

Convolution theorem


25

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)

drrugrfugufuc )-()()(¤)()(∫

∞

∞

•==

Then the function c(u) has Fourier transform C(s) which is the product of Fourier transforms F(s) and G(s)

)()()( sGsFsC •=Very useful attribute in data processing: difficult integral such as convolution can be substituted by simple line ar multiplication of Fourier transforms!

2/2

N.B. Here, the sign ” „ is used to notify the operation of convolution. There is no internatinaly recognized sign.

¤

Excited systems


26

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!

x(t) y(t)Φ

Input signal

Output signal

Excited system

Φ Linear time-invariant operator

System is linear and y(t) is linear combination:

y(t)= Φ{x1(t)+ x2(t)}= Φ{x1(t)} + Φ{x2(t)}= y1(t)+y2(t)

Limitations:

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

There is no difference if input signals are simultaneous or separated in time.


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,


28

x(t) – input signalX(s) – input signal transformy(t) – output signalY(s) – output signal transform

Frequency response: Relationship between the time domain and the frequency domain

h(t) - operator (impulse response) that reproduces input signal to the output one in time domainH(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) mult iplication, and 3) inverse transform than the original convolut ion!!

The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = iω or s = 2πfi


29

Harmonic function as input signalStructural investigations are limiting case in the linear time-invariant systems since an input signal is the harmonic function originated in an electromagnetic field: tietx ω=)(The output signal is a convolution of the impuls response function and the input signal harmonic function

{ } dtethx(t)thtxty ti )()()()()( τωΦ -

-¤

∫

∞

∞

+===

or, after Fourier transforming the impuls response h(t) from the time domaine to the transfer function H(ω) in the frequency domain:

dtethH tiωω )()(∫

∞

∞

+=

-

the output signal can be presented as a linear multiplication of the transfer function and the input signal:

dteHty tiωω)()( =


30

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

00

1

time, t

delta

(t)

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.

Dirac’s functionVery usefull functions in data processing:

=)(xδ { 00 ≠x for 0=+ x for ∞


31

Fourier transform domains in molecular structure analysis

Linear time-invariant system

Time domaine Frequency domaine

Harmonicfunction

Impulsresponse

Transferfunction

FT spectroscopy

Time domaine Frequency domaine

CW-IR Interferogram Complexspectrum

Pulse NMR FID (free –induction decay)

Complex Spectrum

Diffraction Reciprocal space Real space

X-ray F(hkl) Structurefactor

Electron density

Electron Idem Electrostatic potential

Neutron idem Nuclear potential

ωωπ

ω deHth ti)()(∫

∞

∞

21

= dtethH tiω

πω

21

= )()(∫

∞

∞

The use of X-Rays and Fourier transformations for molecular and crystal

structure determination (1912-1914)

� 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

32CEEPUS Summer School in Zagreb July 13, 2010

William Lawrence Bragg 1890-1971

William Henry Brag 1862-1942

..”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 prise1915.

),()( ϕIfSF =

( ) ( ) ))(2exp(∑

1jjj

N

jo

lzkyxhiSICSF ++==

π

( ) ( )j

N

jjc rSiSfSF π2exp)(∑

1=

=

rSS πϕ 2)( =

2* F)()()(I == SFSFS

HOW TO PRESENT THE PHENOMENON OF DIFFRACTION?

Number of diffraction intensities observed, phases LOST


Structure factor (in reciprocal space)

observed

calculated

( ) ***clbkahhklS ++=

Radiusvector in real space

Radiusvector in reciprocal space

dVerSF rSi

r

πρ 2)()(∫

=

( ) ***czbyaxxyzr jjjj ++=

dd –distance between the lattice planes in the real space

A

B

C

Path difference: AB+BC=δAB=δ/2

sinθ=AB/d

sinθ=δ/2d

Bragg’s law:

2dsinθ=nλ

Bragg’s approach to the diffraction condition

Reflection on the planes


Convolution in diffraction experiments


35

(Contribution to phase problem solving)

2* )()( )()(I SFSFSFS ==

dVerdVerS rSi

r

rSi

r

'2-

'

2 )'()()(I∫∫

ππ ρρ=

rru =Substitution:

=+= dVdVerurS uSi

ru

πρρ 2_

)()()(I∫∫

dVeuP uSi

u

π2_

)(∫

)(⇔I rP)S(1st pair of Fourier transformsrepresents the same phenomenon (distribution of interatomic distances) in two different spaces


36

dVruruPr

)()()(∫

+= ρρ

Patterson function: an autoconvoluitionof electron density function found at different points (at u distance) in the same – coordinate system (real space)

=+= dVdVerurSI uSi

ru

πρρ 2_

)()()(∫∫

dVeuP uSi

u

π2_

)(∫

The relation was found:

and by definition

But in practice, since the electron density is unknown, it is calculated as: ∑

)(22

)()(hkl

lzkyhxiehklFxyzP ++= π

In protein crystalography, the intensity F(hkl)2 of diffracted maximums in Patterson function are combined with different additional information

Patterson function

The meaning of Patterson function


37

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

x

ry u

r r’

electron density map ρ(x)

Patterson map P(x)

Real space

x

y

u


38

*2_

*

)()(∫

dVeSFr rSi

V

πρ =

Structure factorin reciprocal space

Electron densityin real (crystal) space

)(⇔)( SFrρ2nd pair of Fourier transformsrepresents the same phenomenon (distribution of electrons) in two different spaces.

The principles of structure compleating and refinement

dVerSF rSi

V

πρ 2)()(∫

=


39

In practice, the maps are calculated as discrete Fourier series,i.e. as FFT (Fast Fourier Transform)

∑

)(2)(I)(hkl

lzkyhxiehklxyzP ++= π

∑

)(2_

)()(hkl

lzkyhxiehklFxyz ++= πρ

)(I⇔r)( SP

)(⇔)( SFrρ

dVruruPr

)()()(∫

+= ρρConnected by convolution:

Connected by multiplication:

2* )()()()(I SFSFSFS ==

Fourier transformation in diffraction experiments

� 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

� Structure solving, completing and refining:1. Patterson map as FT of intensities, 2. electron density map as FT of structure factors, 3. data processing

)( ⇔)( SFrρ

γβα ,,,,, cba , set of hkl

The full use of Reciprocal space in diffraction structure analysis


)(I⇔)( SrP


41

Fourier transformation in IR spectroscopic experiments

Multiplex cos waves by non-dispersive device, i.e. interferometer

IR interferogram in time domain

Spectrum in frequency domainF

Multiplex signal decoding

Albert A. Michelson (1852-1931), Nobel prise 1907


42

Fourier transformation in NMR spectroscopic experiments

FT can be used as:�Pulse NMR signal decoding�Different kinds of signal corrections

FT

Spectrum in frequency domain

Free-induction decay or interference pattern in time domainThe nuclei

shocked into oscillation by means of radiofrequency pulse generator

Single burst of radiofrequency energy

Frequency content of the pulse

F

Richard R. Ernst (1933), Nobel prise1991


43

υυ υπ deSxI xi2

-

)()( ∫+∞

∞

= dxexIS xi υπυ 2-∞

∞-

)()(∫

+

=

Time domain Frequency domaineFT-IR(CW-IR)

FT-NMR(Pulsed NMR)

Response pulse (FID) decreases in time

Response pulse (interferogram) depends on path difference in two arms carrying fixed and movable mirror

SPECTRUM

SPECTRUM

F

F

–x +x

t

FT Spectroscopic signal decoding

Intensity of interferogram

Spectrum


44

Convolution in spectroscopic experiments

Diffraction: convolution has the key role in structure solving:

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.

�Patterson function

Example1. Spectrum filtering byweighting function


45

convolution of spectrum and weighting function transform can be replaiced by a product of interferogram and weighting function

)()(¤)( υυυ corrSWS =

)()()( xIxwxI corr=•

F

CONVOLUTIONF Corrected spectrum

weighting function

F

spectrum

interferogramCorrected intreferogram

weighting function

= (Corrected spectrum)= (spectrum)

MULTIPLICATION

F

Example 2. Truncation


46

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. ApodizationA. Infinitely long cos wave

B. Apodization function

C. Apodized cos wave (multiplied A and B)

D. FT of apodized wave has to be transformed

(πόδι=foot, leg)

Example 3. Apodization (cont)


47

{

+>>

+=

XxXfor

XXforX

x

xa- 0

≤x≤--1)(

Apodization function:

+X

1

x-X Fourier transform of apodizing function:

dxexIS xiff

υπυ 2-∞

∞-)()(

∫

+=

υυυυυυυ dASASS f )-'()()()¤()(∫

∞

∞-

+==

{ }2

2

)(

)(sin)()(

X

XXAxa

πυπυυ ==F

Filtered spectrum:

But using Fourier transforms:

)()()( xaxIxI f = it is easier to calculate

and filtered spectrum can be obtained by simple Fourier transformation:

Example 4. Phase correction


48

Sources of phase error:� error in triggering� start of experiment in t≠0� etc.

Possibble phase correction function: )(xwIts Fourier transform: )(υWRegistered interferogram:

)()()( xwxIxIcorr •=

dxexIS corrcorrxi2-

-

)()( υπυ ∫+∞

∞

=

)(xIcan be corrected by multiplication:

Registered spectrum is difficult to correct using convolution: )()()( υυυ WSS corr ⊗=

But the corrected spectrumis easier to calculate fromcorrected interferogram byFourier transformation:

See Example 1 !


49

Application of convolution:

�Diffraction methods: indirect determination of atomicpositions in crystal (Patterson map)�Spectroscopy methods: interferogram and spectrum corrections

Application of Fourier transformation:

�Diffraction methods: scattering field density map (x-ray/electron density)

�Spectroscopy methods: multiplex signal decoding(interferogram/spectrum)

Résumé


50

Linear respons theory explains why Fourier frequency analysis is applicable to systems excited by broadband electromagnetic radiation.

Conclusion

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

Two domains are communicating by means of Fourier frequency analysis

�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

Fourier synthesis in chemical analysis CEEPUS Zagreb-2010€¦ · Fourier Synthesis in Chemical...

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