Wavelet Analysis in EPR Spectroscopy

Transcription

1 Vol. 108 (2005) ACTA PHYSICA POLONICA A No. 1 Proceeding of the XXI International Meeting on Radio and Microwave Spectrocopy RAMIS 2005, Poznań-Bȩdlewo, Poland, April 24 28, 2005 Wavelet Analyi in EPR Spectrocopy A. Drzewiecki and P.B. Sczaniecki Intitute of Phyic, Univerity of Zielona Góra Szafrana 4a, Zielona Góra, Poland Poibility of application of wavelet analyi in EPR pectrocopy i invetigated. Baic mathematic of the method i ummarized, with emphai on the relation between EPR experiment and phyical interpretation of wavelet tranform. The fact that firt derivative of the Lorentzian function can be ued a a bae wavelet i employed and the pecific rule of the interpretation of the final wavelet ynthei are identified. Analyi of a ample EPR pectrum of X-ray irradiated guanidinium aluminium ulphate hexahydrate crytal i given a an example. Set of parameter reulting from the wavelet analyi i ubequently ued to recontruct/yntheize the original EPR pectrum. The method involve an opportunity of eliminating unwanted component of the multicomponent pectra. Generally, the method may upplement other mean of data analyi, in particular, in magnetic reonance. PACS number: Kf, Rn, Kx 1. Introduction Wavelet analyi i a modern and popular method of data analyi. Application of thi method are urpriingly wide: from eimology through cardiology, it end up with digital image compreion. The baic keyword i multireolution analyi it make it o widely ued. Typical EPR pectrum i a um of component, originating from paramagnetic center of the tudied ample. The line hape of EPR line of particular paramagnetic center i uually Gauian or Lorentzian function, they differ in the reonance field, their amplitude, and line width. We want to obtain magnitude of thee parameter. Multicomponent EPR pectra are uually recorded a derivative of paramagnetic aborption. In order to find reonance field of particular center, we correponding author; A.Drzewiecki@if.uz.zgora.pl (73)

2 74 A. Drzewiecki, P.B. Sczaniecki need to know where the derivative are zero. But in a real pectrum it i difficult to determine thee point, epecially when the line are too cloe to each other, relative to their line width, or if line width ignificantly differ even wore, when their amplitude differ ignificantly. If one of thee fact occur, the typical method of determination of reonance field differentiating the recorded pectrum, and finding maxima of derivative fail. Wavelet analyi of EPR pectrum of guanidinium aluminium ulphate hexahydrate (GASH) crytal irradiated with X-ray i examined and dicued. The pectrum i complex: there are three equal line, originating from SO 3 4 ion [1], but the meaured pectrum exhibit a broad line of large amplitude, which obtruct the analyi of the other line. Wavelet analyi of the pectrum can allow to ditinguih line with different line width, which make poible to extract and remove the broad line from the pectrum and analyze the remaining component, however the removal i neither exact nor automatic. The concluion i that the method i ueful a a upporting tool in the interpretation of thee type of EPR pectra. 2. Baic of wavelet analyi Wavelet analyi of a ignal provide information on both: time and frequency domain. Of coure, the two-dimenional analyi of one-dimenional data et mut be redundant, but the additional information which we obtain i the argument for wide application of thi method. Information which we obtain in the two dimenion i not independent. Localization in time and frequency i limited by Heienberg rule of uncertainty. However, within thi limit we obtain information on both dimenion at once. Alternatively, if we have a ignal, we know exact value of it in a given time, but we know nothing about the frequency. Of coure, we can make the Fourier tranform of thi ignal, and then we will know all about frequencie of the ignal, but we do not know the value of ignal at a given time. Wavelet analyi offer an information on both of thee dimenion imultaneouly [2]. From mathematical point of view, the wavelet tranformation i defined a follow: Ψ(t, ) = f(x) 1 ( ) x t ϕ dx, (2.1) where f(x) i the analyzed function, ϕ i a function called bae wavelet, and t and are parameter, controlling the procee called tranlation and caling, repectively. Interpretation of thi integral i imple: the obtained value i a meaure of imilarity of function f(x) and function ϕ, tranlated t unit to the right on x-axi and then tretched time [3]. Both function f(x) and ϕ mut be quare integrable.

3 Wavelet Analyi in EPR Spectrocopy 75 Proper bae wavelet mut atify the following condition: 1. mean value of bae wavelet mut be equal to zero ϕ(x)dx = 0; (2.2) 2. bae wavelet mut be quare integrable ϕ(x) 2 dx < ; (2.3) 3. the following integral mut have a finite value: C ϕ = ϕ(ω) dω <, (2.4) ω where numerator i the modulu of the Fourier tranform of ϕ. There are phyical argument to adopt Lorentzian or Gauian function a EPR aborption line hape. In the magnetic reonance context, thee function mut be augmented with thee parameter: A amplitude of the line, a multiplicative factor; B 0 reonance field, a a ubtractive factor to magnetic field B; half-width at 1/2 amplitude for the Lorentzian hape (for Gauian function thi definition i imilar, except that half-width mut read at 1/e 1/2.78 amplitude). For Lorentzian function (for Gauian function reaoning i the ame) f(x) L = x 2 ; (2.5) inerting EPR parameter give the following function: ( ) B B0 A f A,B0,(B) L = Af = L 1 + ( ) B B 0 2. (2.6) Lorentzian function i ued in the expreion above in the ame form a the bae wavelet in the wavelet tranform, except multiplication by A. The role of tranlation parameter t i taken by the reonance field B 0 and the role of caling parameter goe to the line width, repectively. The baic problem i that neither Gauian nor Lorentzian function can be bae wavelet they have nonzero mean value. Their firt derivative however, atify all the requirement (Eq. (2.2) ( 2.4)). A mentioned above, EPR pectra are uually recorded a derivative of EPR aborption, which i advantageou for the method. Firt derivative of Lorentzian line ha the form (for derivative of Gauian function the multiplying factor i the ame)

4 76 A. Drzewiecki, P.B. Sczaniecki f A,B 0,(B) L = A f ( ) B B0 L B B 0 = 2 A [ 1 + ( B B 0 ) 2 ] 2. (2.7) Again the form of thi equation i the ame a bae wavelet in wavelet tranform, except for the multiplying factor. Thi lead to key concluion: if recorded pectrum i linear combination (at leat approximately) of derivative of Lorentzian function (Eq. (2.7)), we can eek them by wavelet analyi: et of tranlated and caled bae wavelet i regarded in the calar product, in wavelet tranformation a omething imilar to bae in linear pace. We need to examine the propertie of wavelet tranformation with the elected bae wavelet. Bae wavelet i the baic element ought by the tranformation. With the choice of normalized Lorentzian wavelet ( ) x t ϕ = 4 π [ x t 1 + ( x t the tranformation ha the form Ψ(t, ) = f A,B 0,(x) L ( t B0 ) 2 ] 2, (2.8) ( ) 1 x t ϕ dx ) 2 = 4A π ( + ) 3 [ ( ) ] 2 3. (2.9) t B The propertie of thi tranformation are: with repect to the variable t, wavelet tranformation of Lorentzian line ha the hape of econd derivative of Lorentzian function (Fig. 1); treated a a function of two variable (Eq. (2.9)) ha one maximum at (t = B 0, = ). The coordinate of the maximum yield two of the three baic parameter of the analyzed pectral line; knowing the coordinate of the maximum and it magnitude, one may alo compute the amplitude of the line from the following formula: A = 2 Ψ(t, ). (2.10) π t=b0,= propertie of the tranformation with repect to variable caue a uppreion of thee component whoe line width ignificantly differ from the local value of. Thi mechanim may help to filter a line of a pecified width; tranformation of a ingle line conit of three region: central poitive part that include the global maximum and two negative part on left and right ide, their eparation i a linear function of.

5 Wavelet Analyi in EPR Spectrocopy 77 Fig. 1. Shape of wavelet tranformation with repect to the variable t. The horizontal line mark zero level. Fig. 2. Reult of wavelet analyi of a ample Lorentzian derivative obtained with numerical program Analizator *. Horizontal and vertical axi denote t and, repectively. Each line in the complex pectrum generate additive ingredient in the form of above function to ummary wavelet analyi. In thi particular picture the pattern from left to right correpond to negative-poitive-negative value, reflecting the hape of the function in Fig Wavelet analyi an example Guanidinium aluminium ulphate hexahydrate i a ferroelectric with pace group C 2 3v (P 31m). The crytal formula i C(NH 2 ) 3 Al(H 2 O) 6 (SO 4 ) 2 : three nearly planar guanidinium group are andwiched between the hexaquo metal octahedra. Expoing the crytal to X-ray K α radiation for 2 hour produce paramagnetic, aniotropic center whoe EPR will be ued a an example of the application of the wavelet analyi. Three overlapping component (Fig. 3) can be ditinguihed, all in the vicinity of free radical g-value [4]. Wavelet analyi of thi EPR pectrum (Fig. 3) i preented below (Fig. 4a). The red-colored region repreent the poitive magnitude and the blue-colored region repreent the negative value of the analyi (Eq. (2.9)). Horizontal cro-ection of the above picture reveal that the analyi ha a maximum at (t 1, 1 ). Large value of 1 give evidence of a broad line. On the The program, written in C++, i available from the primary author upon requet.

6 78 A. Drzewiecki, P.B. Sczaniecki Fig. 3. Recorded EPR pectrum of X-ray irradiated GASH crytal. Fig. 4. (a) Wavelet analyi of EPR pectrum (Fig. 3). (b) Wavelet analyi of pectrum (Fig. 3) after eliminating the broad line. Horizontal and vertical axi denote t and, repectively. cale of magnetic field t 1 equal B 1. At the bottom part of Fig. 4a additional feature with mall line width are viible. We can eliminate the broad line, uing the parameter obtained previouly (t 1, 1 ), and the amplitude calculated from the formula (Eq. (2.10)). Thu, the filtered pectrum i obtained and hown in Fig. 4b. Analyi of it how two relatively narrow line, whoe parameter are (t 2, 2 ) and (t 3, 3 ), with A 2 and A 3, correpondingly. The et of the parameter reulting from the analyi i preented in Table. An attempt ha been made to yntheize the original EPR pectrum (Fig. 3) from the obtained parameter. The reult are diplayed in Fig. 5. The blue line, which i the final reult of the analyi, can be compared to the plot in Fig. 3. It i the um of the three extracted component: orange line #1, green line #2 and magenta #3. One may conclude that the wavelet analyi i a complementary tool for pectra interpretation; the yntheized pectrum reemble the experimental ob-

7 Wavelet Analyi in EPR Spectrocopy 79 TABLE Parameter of the wavelet-detected EPR line. Component A t Fig. 5. Synthei of EPR pectrum with the parameter hown in Table. ervation. A an additional utility one ha the option of earching and eliminating unwanted feature from the experimental data. Reference [1] B.J.B. Schein, E.C. Lingafelter, J.M. Stewart, J. Chem. Phy. 47, 5138 (1967). [2] D.B. Percival, A.T. Walden, Wavelet Method for Time Serie Analyi, Cambridge Univerity Pre, Cambridge [3] J.T. Bia laiewicz, Wavelet and Approximation, Wydawnictwo Naukowo- -Techniczne, Warzawa 2004, p. 66 (in Polih). [4] P.B. Sczaniecki, B. Jakubazek, L. Szczepańka, in: Magnetic Reonance and Related Phenomena, Vol. II, Extended Abtract of the Joint 29th AMPERE 13th ISMAR Int. Conf., Ed. D. Zieow, W. Lubitz, F. Lendzian, Techniche Univerität Berlin, Berlin 1998, p. 981.