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Amplitude modulated transversal and longitudinal EPR spectroscopy
Solid State Communications, Vol. 52, No. 4, pp. 4 2 3 - 4 2 5 , 1984. Printed in Great Britain.
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
AMPLITUDE MODULATED TRANSVERSAL AND LONGITUDINAL EPR SPECTROSCOPY M. Giordano, M. Martinelli and L. Pardi Dipartimento di Fisica, Universit/t di Pisa, Gruppo Nazionale di Struttura della Materia del CNR and Centro Interuniversitario di Struttura della Materia, Pisa, Italy S. Santucci Dipartimento di Fisica, Universit~t di Perugia, Gruppo Nazionale di Struttura della Materia del CNR and Centro Interuniversitario di Struttura della Materia, Perugia Italy and C. Umeton Dipartimento di Fisica, Universith della Calabria, Cosenza, Italy (Received lO June 1984 by F Bassani) The amplitude modulated EPR spectroscopy is analyzed both in the time and the frequency domain. The results of numerical calculations and analytical approximate treatments indicate that the signal lineshape is differently affected by relaxation mechanisms when transversal or longitudinal detection is used in spectroscopies with variable frequencies of modulation. Measurements of longitudinally detected electron-spin double resonance obtained in dependence on the frequency of modulation confirm the lineshape expected by the theoretical analysis.
1. INTRODUCTION SINGLE OR MULTIPLE continuous wave irradiation techniques have been largely applied in electron paramagnetic resonance (EPR) spectroscopy. Recently, time-dependent spectroscopy techniques have attracted a growing interest, since they can be used in studying special phenomena or in obtaining data which could not be measured with other procedures. Among the most successful fields of applications, we mention the electron-spin echo [1 ], CIDEP [2], amplitude modulated spectroscopy (AMS) [3], time dependent longitudinal effects [4]. The theoretical treatment of all the above fields of research requires solving non-stationary master equations, for which several procedures have been devised using electronic processors. We consider here, in particular, the AMS techniques. They have been widely used both in order to measure spin-lattice relaxation times [5] and as an alternative procedure [3] in place of the standard field modulated EPR spectroscopy. In the present letter we give a treatment of the AMS techniques in EPR spectroscopy introducing an interpretative scheme for the experimental results, which can be applied directly either to the time evolution of the signal or to the frequency response of the studied systems. The flexibility of our procedure allows to take advantage from both points of view, and to obtain provisions on the
behaviour of physical systems in different conditions. "Ad hoc" measurements are also presented to check the theoretical analysis. In particular in the case of AMS techniques applied with variable modulation frequency, we show that the lineshape of the signal varies with the modulation frequency in different ways depending on whether the longitudinal or the transversal components of the magnetization are observed. This effect is both theoretically derived and experimentally measured. It is also worth noting here that calculating the system response implies a noticeable difficulty when nearly unit index modulation is used. In this case no perturbative approach is suitable. Our treatment shows how to deal with such a condition.
423
2. THEORETICAL In the following, we are concerned with an electron spin system acted on by a static magnetic field, and irradiated by an amplitude modulated microwave. No restriction need to be imposed either to the index or to the frequency of modulation. The microwave magnetic field is H1 = H(t) cos w t where co ~ wo (Larmor frequency) and H(t) = H + h cos ~ t ; ~ being the modulation frequency. The response of the spin system varies in time, and its shape depends on ~2 and on the relaxation time values. Moreover, different effects are expected if
424
TRANSVERSAL AND LONGITUDINAL EPR SPECTROSCOPY
transversal or longitudinal components of the magnetization are observed.
Vol. 52, No. 4
longitudinal calculated signals are matched by simple functions. In the case of the terms oscillating at the lowest frequencies, the signal amplitudes are given by:
2.1. Time dependent analysis To obtain the time response for the problem in the general case we have implemented a computer program in the CSMP Ill-IBM language. This language enables a continuous process to be simulated [4] and supplies many useful signal elaborations, such as integration, differentiation, transforms and so on, with very simple instructions. Statistical aspects of the problem are treated with the standard methods of the density operator. By introducing the second quantization formalism for the radiofrequency field H~(t), we may re-write the Hamiltonian without explicit time dependence but for the wave amplitude H(t). When a 1/2 spin system is considered, all operators are represented on a basis of two states only. We write the p density operator as p = pO + D, where/90 is the density operator at thermodynamic equilibrium and D represents the part of p affected by relaxation mechanisms. By labeling the states of the 1/2 spin basis with the numbers 1 and 2, we obtain that the longitudinal component of the spin magnetization is proportional to pO, 1 + D1, 1, while the transversal one is proportional to D1, 2. In our case the D~, 1 and D1, 2 terms oscillate at the frequency ~2 and its harmonics. We solve numerically the master equation for D1, 1 and D~, 2, without limitation as to the radio-frequency field intensity and as to the index and the frequency of modulation. The time dependent signals obtained as numerical solutions from our computer program, can be Fourier-transformed in order to evaluate each harmonics within the signal itself. In such way we are able to reproduce whatever physical condition under amplitude modulation experimental technique. The case of 100% amplitude modulation is of particular interest since the different behaviour of the transversal and longitudinal component of the magnetization is easily interpreted both in the time and in the frequency domains. In such a case we have/7 = 0 and Hi(t) is given by: Hi(t) = h cos ~2t cos cot = h/2[cos (co + ~2)t + c o s (co -
s2)t].
By applying the above procedure of analysis at the time dependent response we obtain two main results: (i) the transversal component of magnetization oscillates at the modulation frequency ~2 and its harmonics, while the longitudinal component oscillates at the frequency 2~2 and its harmonics. (ii) at resonance (co ~ COo), and far from saturation, the variation with ~ of both the transversal and the
IM~I)[ cc
IM(2) z
T2 1 + ~22T~ '
(1)
Tl T2 (1 + 4~22T?)'/2(1 + ~22T~)'/2 "
(2)
Equations (1) and (2) are obtained assuming H1 parallel to the x-axis. The superscripts (1) and (2) refer to the harmonics of the ~2 frequency which each component oscillates. It is seen directly from equations (1) and (2) that IM~I)I depends on the transversal relaxation time 7"2 only, both transversal/'2 and longitudinal T1 relaxation times appear in kl'/z(2)[.
2.2. Frequency domain analysis We consider now the response of the same physical system as the one discussed above and we analyze it in the frequency domain. The system is irradiated by 100% amplitude modulated wave. By assuming a linear response for the system, this can be described as if it were irradiated by two independent waves with equal amplitudes, and with frequencies co + ffZ and co -respectively. Within this assumption, the transversally detected signal - in the first harmonics - may be derived by summing the responses (for varying ~ ) to each one of the two irradiating waves. These two responses can be analytically calculated in an independent way and, in resonance conditions (co = coo), their sum gives just the relation (1). As far as the longitudinal response is concerned, the signal can be calculated by the standard procedures introduced by the Longitudinally Detected Electron Spin Resonance (LODESR) spectroscopy [8]. We simply recall that in this procedure the longitudinal component of the magnetization of an electron spin system irradiated with two transversal waves r and s with angular frequencies cot and cos respectively, both near to the resonance frequency coo, oscillates at A 2 = ] 6 0 r - - COs[ frequency and its harmonics. At the lowest interaction order this phenomenon corresponds to a non linear process, since it represents the absorption of one r wave photon and the emission of one s wave photon. In the present context, these two waves are represented by the two sidebands. The first harmonics for the longitudinal signal occurs at A2 = 2~2. The general equations of the LODESR signal [8], after some algebraic manipulations, give analytically the expression of IMP2) I coinciding with that of equation (2) under the above hypotheses and by neglecting all the processes involving more than two photons.
Vol. 52, No. 4
TRANSVERSAL AND LONGITUDINAL EPR SPECTROSCOPY
~. SLO D
LN T
,i ,2\ n"
,.~ (105Hz)
5 Fig. 1. First harmonics signals in LODESR frequencyswept experiment at room temperature (RT) and liquid nitrogen temperature (LNT). Dotted lines represent the approximate analytical expression, solid lines the numerical calculations. Triangles are experimental points at RT and circles at LNT. The microwave fields are in both cases H f =HI" = 4 x 10 -2 G.
temperature (LNT). The T2 value in both cases is 7.4 x 10 -7 secrad -1. In Fig. 1 the measured lMz(2)l values are 'reported vs the frequency ~2 in the same arbitrary units for RT and LNT. Since in the experiment the microwave magnetic field amplitude is fixed, the saturation factor assumes two different values at RT and LNT (S = 0.08 and S = 0.14, respectively). In Fig. 1 the values of 13'/}2)1 given by equation (2) are also represented. There is a fair agreement between experiments and the theoretically expected trend. In fact the line intensity grows while the linewidth decreases when 7"1 increases. Some discrepancies with experimental results arise since equation (2) does not contain saturation effects, which actually are not completely negligible, particularly at LNT. In the figure we also report (solid lines) the results obtained by numerical calculation of [M}2)I performed from general LODESR equations. In this way processes involving up to six photons are taken into account and moreover saturation effects are automatically included. The comparison of experimental results with these numerical solutions exhibits a good agreement as can be seen in Fig. 1. In conclusion, we have shown how the analysis of the lineshape of an amplitude modulated EPR spectrum can give selective informations about the two relaxation times according to the component of the magnetization observed by the detection system. REFERENCES 1.
3. EXPERIMENTAL AND CONCLUSIONS As discussed in Section 2.2, the dependence of IMz~2)lon ~ may be experimentally observed if frequency dependent LODESR experiments are performed. For this purpose we have partially modified our LODESR spectrometer, taking particular care over the equalization of the detection system response vs the variable frequency. In our spin system, we used the electrolyticaUy obtained oxypyrrol radical [9]. This system is well suited for our purposes since - due to the involved relaxation mechanisms - only the longitudinal processes are affected when the sample temperature is varied. Therefore, for the longitudinal relaxation time we have T1 = 8.5 × 10 -7 sec rad -1 at room temperature (RT) and T1 = 1.5 x 1 0 - 6 secrad -1 at liquid nitrogen
425
2. 3. 4. 5. 6. 7. 8. 9.
L. Kevan & R.N. Schwartz, Time Domain Electron Spin Resonance. John Wiley & Sons, New York (1979). L.T. Muus, P.W. Atkins, K.A. McLauchlan & J.B. Pedersen, Chemically Induced Magnetic Polarization. Reidel Publishing Company, Boston (1977). J.C. Gourdon, P. Lopez, C. Rey & J. Pescia, C.R. Aead. Sei. Paris 271,288 (1970); T. Islam & I. Miyagawa, J. Mag. Resonance 51,383 (1983). M. Giordano, M. Martinelli, S. Santucci & C. Umeton, J. Magn. Resonance 47, 133 (1982). J. Pescia, Ann. Phys. (Paris) 10, 389 (1965). M. Matti Maricq & J.S. Waugh, J. Chem. Phys. 70, 3300 (1979). A. Di Giacomo & S. Santucci, Nuovo Cimento !t63,407 (1969). M. Martinelli, L. Pardi, C. Pinzino & S. Santucci, Phys. Rev. B16, 164 (1977). G. Dascola, C. Giori, V. Varacca & L. Chierici, CR. Acad. Sci. Paris 262, 1917 (1966).
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
AMPLITUDE MODULATED TRANSVERSAL AND LONGITUDINAL EPR SPECTROSCOPY M. Giordano, M. Martinelli and L. Pardi Dipartimento di Fisica, Universit/t di Pisa, Gruppo Nazionale di Struttura della Materia del CNR and Centro Interuniversitario di Struttura della Materia, Pisa, Italy S. Santucci Dipartimento di Fisica, Universit~t di Perugia, Gruppo Nazionale di Struttura della Materia del CNR and Centro Interuniversitario di Struttura della Materia, Perugia Italy and C. Umeton Dipartimento di Fisica, Universith della Calabria, Cosenza, Italy (Received lO June 1984 by F Bassani) The amplitude modulated EPR spectroscopy is analyzed both in the time and the frequency domain. The results of numerical calculations and analytical approximate treatments indicate that the signal lineshape is differently affected by relaxation mechanisms when transversal or longitudinal detection is used in spectroscopies with variable frequencies of modulation. Measurements of longitudinally detected electron-spin double resonance obtained in dependence on the frequency of modulation confirm the lineshape expected by the theoretical analysis.
1. INTRODUCTION SINGLE OR MULTIPLE continuous wave irradiation techniques have been largely applied in electron paramagnetic resonance (EPR) spectroscopy. Recently, time-dependent spectroscopy techniques have attracted a growing interest, since they can be used in studying special phenomena or in obtaining data which could not be measured with other procedures. Among the most successful fields of applications, we mention the electron-spin echo [1 ], CIDEP [2], amplitude modulated spectroscopy (AMS) [3], time dependent longitudinal effects [4]. The theoretical treatment of all the above fields of research requires solving non-stationary master equations, for which several procedures have been devised using electronic processors. We consider here, in particular, the AMS techniques. They have been widely used both in order to measure spin-lattice relaxation times [5] and as an alternative procedure [3] in place of the standard field modulated EPR spectroscopy. In the present letter we give a treatment of the AMS techniques in EPR spectroscopy introducing an interpretative scheme for the experimental results, which can be applied directly either to the time evolution of the signal or to the frequency response of the studied systems. The flexibility of our procedure allows to take advantage from both points of view, and to obtain provisions on the
behaviour of physical systems in different conditions. "Ad hoc" measurements are also presented to check the theoretical analysis. In particular in the case of AMS techniques applied with variable modulation frequency, we show that the lineshape of the signal varies with the modulation frequency in different ways depending on whether the longitudinal or the transversal components of the magnetization are observed. This effect is both theoretically derived and experimentally measured. It is also worth noting here that calculating the system response implies a noticeable difficulty when nearly unit index modulation is used. In this case no perturbative approach is suitable. Our treatment shows how to deal with such a condition.
423
2. THEORETICAL In the following, we are concerned with an electron spin system acted on by a static magnetic field, and irradiated by an amplitude modulated microwave. No restriction need to be imposed either to the index or to the frequency of modulation. The microwave magnetic field is H1 = H(t) cos w t where co ~ wo (Larmor frequency) and H(t) = H + h cos ~ t ; ~ being the modulation frequency. The response of the spin system varies in time, and its shape depends on ~2 and on the relaxation time values. Moreover, different effects are expected if
424
TRANSVERSAL AND LONGITUDINAL EPR SPECTROSCOPY
transversal or longitudinal components of the magnetization are observed.
Vol. 52, No. 4
longitudinal calculated signals are matched by simple functions. In the case of the terms oscillating at the lowest frequencies, the signal amplitudes are given by:
2.1. Time dependent analysis To obtain the time response for the problem in the general case we have implemented a computer program in the CSMP Ill-IBM language. This language enables a continuous process to be simulated [4] and supplies many useful signal elaborations, such as integration, differentiation, transforms and so on, with very simple instructions. Statistical aspects of the problem are treated with the standard methods of the density operator. By introducing the second quantization formalism for the radiofrequency field H~(t), we may re-write the Hamiltonian without explicit time dependence but for the wave amplitude H(t). When a 1/2 spin system is considered, all operators are represented on a basis of two states only. We write the p density operator as p = pO + D, where/90 is the density operator at thermodynamic equilibrium and D represents the part of p affected by relaxation mechanisms. By labeling the states of the 1/2 spin basis with the numbers 1 and 2, we obtain that the longitudinal component of the spin magnetization is proportional to pO, 1 + D1, 1, while the transversal one is proportional to D1, 2. In our case the D~, 1 and D1, 2 terms oscillate at the frequency ~2 and its harmonics. We solve numerically the master equation for D1, 1 and D~, 2, without limitation as to the radio-frequency field intensity and as to the index and the frequency of modulation. The time dependent signals obtained as numerical solutions from our computer program, can be Fourier-transformed in order to evaluate each harmonics within the signal itself. In such way we are able to reproduce whatever physical condition under amplitude modulation experimental technique. The case of 100% amplitude modulation is of particular interest since the different behaviour of the transversal and longitudinal component of the magnetization is easily interpreted both in the time and in the frequency domains. In such a case we have/7 = 0 and Hi(t) is given by: Hi(t) = h cos ~2t cos cot = h/2[cos (co + ~2)t + c o s (co -
s2)t].
By applying the above procedure of analysis at the time dependent response we obtain two main results: (i) the transversal component of magnetization oscillates at the modulation frequency ~2 and its harmonics, while the longitudinal component oscillates at the frequency 2~2 and its harmonics. (ii) at resonance (co ~ COo), and far from saturation, the variation with ~ of both the transversal and the
IM~I)[ cc
IM(2) z
T2 1 + ~22T~ '
(1)
Tl T2 (1 + 4~22T?)'/2(1 + ~22T~)'/2 "
(2)
Equations (1) and (2) are obtained assuming H1 parallel to the x-axis. The superscripts (1) and (2) refer to the harmonics of the ~2 frequency which each component oscillates. It is seen directly from equations (1) and (2) that IM~I)I depends on the transversal relaxation time 7"2 only, both transversal/'2 and longitudinal T1 relaxation times appear in kl'/z(2)[.
2.2. Frequency domain analysis We consider now the response of the same physical system as the one discussed above and we analyze it in the frequency domain. The system is irradiated by 100% amplitude modulated wave. By assuming a linear response for the system, this can be described as if it were irradiated by two independent waves with equal amplitudes, and with frequencies co + ffZ and co -respectively. Within this assumption, the transversally detected signal - in the first harmonics - may be derived by summing the responses (for varying ~ ) to each one of the two irradiating waves. These two responses can be analytically calculated in an independent way and, in resonance conditions (co = coo), their sum gives just the relation (1). As far as the longitudinal response is concerned, the signal can be calculated by the standard procedures introduced by the Longitudinally Detected Electron Spin Resonance (LODESR) spectroscopy [8]. We simply recall that in this procedure the longitudinal component of the magnetization of an electron spin system irradiated with two transversal waves r and s with angular frequencies cot and cos respectively, both near to the resonance frequency coo, oscillates at A 2 = ] 6 0 r - - COs[ frequency and its harmonics. At the lowest interaction order this phenomenon corresponds to a non linear process, since it represents the absorption of one r wave photon and the emission of one s wave photon. In the present context, these two waves are represented by the two sidebands. The first harmonics for the longitudinal signal occurs at A2 = 2~2. The general equations of the LODESR signal [8], after some algebraic manipulations, give analytically the expression of IMP2) I coinciding with that of equation (2) under the above hypotheses and by neglecting all the processes involving more than two photons.
Vol. 52, No. 4
TRANSVERSAL AND LONGITUDINAL EPR SPECTROSCOPY
~. SLO D
LN T
,i ,2\ n"
,.~ (105Hz)
5 Fig. 1. First harmonics signals in LODESR frequencyswept experiment at room temperature (RT) and liquid nitrogen temperature (LNT). Dotted lines represent the approximate analytical expression, solid lines the numerical calculations. Triangles are experimental points at RT and circles at LNT. The microwave fields are in both cases H f =HI" = 4 x 10 -2 G.
temperature (LNT). The T2 value in both cases is 7.4 x 10 -7 secrad -1. In Fig. 1 the measured lMz(2)l values are 'reported vs the frequency ~2 in the same arbitrary units for RT and LNT. Since in the experiment the microwave magnetic field amplitude is fixed, the saturation factor assumes two different values at RT and LNT (S = 0.08 and S = 0.14, respectively). In Fig. 1 the values of 13'/}2)1 given by equation (2) are also represented. There is a fair agreement between experiments and the theoretically expected trend. In fact the line intensity grows while the linewidth decreases when 7"1 increases. Some discrepancies with experimental results arise since equation (2) does not contain saturation effects, which actually are not completely negligible, particularly at LNT. In the figure we also report (solid lines) the results obtained by numerical calculation of [M}2)I performed from general LODESR equations. In this way processes involving up to six photons are taken into account and moreover saturation effects are automatically included. The comparison of experimental results with these numerical solutions exhibits a good agreement as can be seen in Fig. 1. In conclusion, we have shown how the analysis of the lineshape of an amplitude modulated EPR spectrum can give selective informations about the two relaxation times according to the component of the magnetization observed by the detection system. REFERENCES 1.
3. EXPERIMENTAL AND CONCLUSIONS As discussed in Section 2.2, the dependence of IMz~2)lon ~ may be experimentally observed if frequency dependent LODESR experiments are performed. For this purpose we have partially modified our LODESR spectrometer, taking particular care over the equalization of the detection system response vs the variable frequency. In our spin system, we used the electrolyticaUy obtained oxypyrrol radical [9]. This system is well suited for our purposes since - due to the involved relaxation mechanisms - only the longitudinal processes are affected when the sample temperature is varied. Therefore, for the longitudinal relaxation time we have T1 = 8.5 × 10 -7 sec rad -1 at room temperature (RT) and T1 = 1.5 x 1 0 - 6 secrad -1 at liquid nitrogen
425
2. 3. 4. 5. 6. 7. 8. 9.
L. Kevan & R.N. Schwartz, Time Domain Electron Spin Resonance. John Wiley & Sons, New York (1979). L.T. Muus, P.W. Atkins, K.A. McLauchlan & J.B. Pedersen, Chemically Induced Magnetic Polarization. Reidel Publishing Company, Boston (1977). J.C. Gourdon, P. Lopez, C. Rey & J. Pescia, C.R. Aead. Sei. Paris 271,288 (1970); T. Islam & I. Miyagawa, J. Mag. Resonance 51,383 (1983). M. Giordano, M. Martinelli, S. Santucci & C. Umeton, J. Magn. Resonance 47, 133 (1982). J. Pescia, Ann. Phys. (Paris) 10, 389 (1965). M. Matti Maricq & J.S. Waugh, J. Chem. Phys. 70, 3300 (1979). A. Di Giacomo & S. Santucci, Nuovo Cimento !t63,407 (1969). M. Martinelli, L. Pardi, C. Pinzino & S. Santucci, Phys. Rev. B16, 164 (1977). G. Dascola, C. Giori, V. Varacca & L. Chierici, CR. Acad. Sci. Paris 262, 1917 (1966).