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Enhanced accuracy of the microwave field strength measurement in a CW-EPR by pulsed modulation technique
Journal of Magnetic Resonance 287 (2018) 123–127
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Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Enhanced accuracy of the microwave field strength measurement in a CW-EPR by pulsed modulation technique B. Rakvin ⇑, D. Caric´, M. Kveder Ruder Boškovic´ Institute, Division of Physical Chemistry, Bijenicˇka 54, Zagreb, Croatia
a r t i c l e
i n f o
Article history: Received 11 December 2017 Revised 9 January 2018 Accepted 10 January 2018 Available online 11 January 2018 Keywords: EPR spectroscopy Microwave field strength Modulation spectrum Amplitude modulation EPR Bloch-Siegert shift
a b s t r a c t The microwave magnetic field strength, B1, in the cavity of a conventional continuous wave electron paramagnetic resonance, CW-EPR, spectrometer was measured by employing modulation sidebands, MS, in the EPR spectrum. MS spectrum in CW-EPR is produced by applying the modulation frequency, xrf, which exceeds the linewidth, dB, given in frequency units. An amplitude-modulated CW-EPR, AM-CW-EPR, was selected as detection method. Theoretical description of AM-CW-EPR spectrum was modified by adding Bloch-Siegert-like shift obtained by taking into account the cumulative effect of the non-resonant interactions between the driving fields and the spin system. This approach enables to enhance the precision of B1 measurement. In order to increase the sensitivity of the method when saturation effects, due to higher intensity of B1, decrease the resolution of AM-CW-EPR spectrum, detection at the second harmonic of CW-EPR has been employed. Ó 2018 Elsevier Inc. All rights reserved.
1. Introduction The microwave magnetic field strength, B1, as well as closely related Rabi frequency (x1 = ceB1) of a spin system in the microwave field, MW, are essential parameters for all quantitative measurements in EPR spectroscopy. In recent years studies of spin system containing very long relaxation times (transverse, T2 and longitudinal, T1) are of interest due to their potential application in quantum information technologies. One of the most often addressed model cases in these studies is the resonant interaction between the bichromatic field (transverse MW field and longitudinal radiofrequency (RF) field) and two level spin system (spin qubit) experiencing the Zeeman magnetic field. Early theoretical study of modulation effect in EPR spectroscopy was based on semi-classical approach by applying modified Bloch equations [1]. However, recently the description was revised and modulation effects in the continuous wave EPR, CW-EPR, spectroscopy explained by introducing multi-photon transitions [2,3]. By using quantized radiation fields it was shown [2,3] that modulation sidebands, MS, appearing in the EPR spectrum when the modulation frequency, xrf, used in CW-EPR exceeds the linewidth, dB, can be described by the multi-photon transitions. In these transitions one MW r+ photon is absorbed from the MW radiation field and
⇑ Corresponding author. E-mail address: [email protected] (B. Rakvin). https://doi.org/10.1016/j.jmr.2018.01.006 1090-7807/Ó 2018 Elsevier Inc. All rights reserved.
an arbitrary number k of RF p photons are absorbed from or emitted to the modulation RF field (r+ + k p). In the present consideration among the various different methods [4–11] for estimation of B1, the experimental method with pulsed modulation detection proposed several decades ago [12] will be discussed and revised within recently suggested multiple photon description of CW-EPR spectra. The method is based on the measurements of splitting between the first sideband signals [10,11], d1, under the small modulation index z = 2x2/xrf (2x2 corresponds to the amplitude of RF field), (z 1) [12]
2 x2rf d1 ¼ 2 B21 2 ce
ð1Þ
One can measure d1 for several values of xrf, and from the intercept on the abscissa of the plot (d1/2)2 versus xrf2 the value of B1 can be obtained in accord with Eq. (1). This approach shows several advantages in comparison with other methods [4,5,8,9]. As was shown earlier the method [12] requires no additional calibration procedure and yields B1 values directly in a wide range of microwave power. Moreover, the pulsed modulation technique can be performed by employing standard audiofrequency lock-in amplifier for detection of the dc signal component rather than the first harmonic component of a sideband spectrum. It could be noted that detection of the dc signal of sidebands spectrum is possible with low frequency pulse modulation (within range of 0–100 kHz as expected for lock-in amplifier included in a standard CW-EPR spectrometer) in
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comparison with the first harmonic detection [10,11] where significantly large xrf is usually required (xrf 1 MHz) for producing sideband’s splitting and lock-in detection at the same frequency [2]. However, there is some limitation in this method due to approximation in theoretical description, which is based on the semi-classical rotation frame model [2]. The sideband’s splitting taken for close sideband’s resonance (where condition (x1/xrf 1) is not preserved and transforms towards the condition (x1/xrf < 1)) deviate from the straight line in the plot d21 versus xrf2 [12]. Therefore, these experimental points were avoided in the process of evaluation of the B1. 2. Effect of Bloch-Siegert-like shift on MS spectrum 2.1. Amplitude-modulated CW-EPR with Bloch-Siegert-like shift The experimental method of pulsed modulation detection employed earlier generally coincides with more recently suggested method of amplitude-modulated CW-EPR, AM-CW-EPR [13,14]. Thus, in order to simulate experimental modulation spectra detected previously [12] and to evaluate more accurately the corresponding B1, the theoretical description of the AM-CWEPR spectrum for an individual homogeneous line is required. Presently suggested theoretical description [13,14] includes B1 exclusively in the saturation term, while the splitting between sidebands is lacking the B1 dependence and it is proportional only to the multiple of modulation frequency (kxrf). Therefore, here presented study improves this description by introducing Bloch-Siegert-like shift, Dk, of the energy levels obtained from the expanded Hamiltonian for resonant multiple photon transitions [2,3,15,16]. Following recent theoretical description of spin dynamics which can be treated with frequency parameters in the range of CW-EPR spectroscopy [16], it is convenient to introduce two different regimes of the spin dynamics such as: ‘‘weak modulation near the Rabi resonance” and ‘‘strong and fast modulation” with the corresponding values of the applied RF modulation (x2 x1 xrf) and (x2 > x1 xrf), respectively. For each of these regimes the corresponding Dk was suggested and calculated. In accord with the selection of RF for the ‘‘strong and fast modulation” regime Dk was calculated [15,16] up to the second order contribution in small parameter x1/xrf. By employing these results and an additional term of Dk for the evaluation a homogenous EPR line [16] the new description of AM-CW-EPR spectrum [13,14] is here suggested 1 X
P¼
"
x1 T 2 J2k ðzÞ 2
k¼1
Dk ¼
1 þ x21 T 1 T 2 J 2k ðzÞ þ ðXs kxrf þ Dk Þ T 22 # x1 T 2 J2k ð0Þ 2
1 þ x21 T 1 T 2 J 2k ð0Þ þ ðXs kxrf þ Dk Þ T 22
ð2Þ
2 X 2 x21 x1 J l ðzÞ þ 4xmw 2xrf l–k ðk lÞ
where z = 2x2/xrf, Jk(z) is the Bessel function of the first kind and order k, Xs = xs xmw and xs = ceB0. For weak MW fields, as commonly used in CW-EPR the first term in Dk is small and can be neglected. It can be shown that Dk = Dk and there is no shift for D0 = 0 (k = 0, centerband) while the largest contribution to the splitting is expected between the first sidebands in comparison with the splitting between the second and higher order of sidebands. From the Eq. (2) one can easily evaluate the new corrected splitting between sidebands peaks as:
ce dk 2
¼ kxrf Dk
ð3Þ
In order to compare Eqs. (1) and (3), Eq. (3) can be transformed in the form of Eq. (1)
!2 2 X 2 2 X J 2 ðzÞ x2rf d1 1 x1 J l ðzÞ 2 l ¼ 2 B1 þ ð1 lÞ c2e 2xrf ð1 lÞ 2 ce l–1 l–1
ð4Þ
The obtained result approaches the result derived from the rotation frame model given in Eq. (1) when assuming conditions for strong and fast modulation (z 1, and x1/xrf 1). Under these conditions and by taking the leading terms in the Bessel sum approaching unit value the Eq. (4) is further simplified
2 2 x2rf d1 z2 1 2 2 4 ce
1þ
D21
x2rf
!
1
z2 4
2 B21
ð5Þ
The obtained relation increases accuracy of a description of the intercept on the abscissa of the plot (d1/2)2 versus xrf2 in comparison with the earlier description given by Eq. (1). The effect of these corrections can be easily followed by presenting the data using scaled plot (d1/2)2(1 z2/4)2 versus (xrf/ce)2(1 z2/4)2 (1 + D21/xrf2). This approach includes the change of z and D1 parameters as is expected for measurements at different xrf values. Additionally, the similar analyses can be provided for the second spin dynamics regime ‘‘weak modulation near the Rabi resonance” and the corrected separation between the sidebands peaks is expected as follows [16]
2 x2rf Dð1Þ2 d1 ¼ 2 1 B21 2 ce xrf "
Dð1Þ ¼
x22 sin2 h X 1 2 ðJ ðaÞ þ J n ðaÞJ n2 ðaÞÞ 2xrf n n n–0 # 1 ðJ 2n ðaÞ þ J n ðaÞJ nþ2 ðaÞÞ nþ2 n–2
þ
X
ð6Þ
where a = (2x2/xrf) cos h, sin h = x1/(x21 + Xs2)1/2 and cos h = Xs/ (x21 + Xs2)1/2. In this dynamics regime D(1) is calculated up to the second order contribution by assuming (x2/xrf) sin h 1 as a small parameter. The advantage of this theoretical result is that one can describe the splitting between sidebands when RF modulation is in the vicinity of Rabi resonance (x1 xrf) [16,17]. It is expected that d1 exhibits minimum at resonance condition (x1 = xrf) and amplitudes of low-field and high-field sideband are not equal as was demonstrated in the time resolved EPR experiment [17]. Thus, the choice of the appropriate relation to express measured data in terms of d21 versus xrf2 at constant microwave power depends on the characteristic relaxation parameters T1 and T2 of spin system and on the applied regime of spin dynamics (x2, xrf and x1). One expects that for more reliable measurements of d1 it is convenient to use larger values of, d1 2xrf than the smaller values of d1 near the low limit, d1 dB. This requires measurements in the fast motional regime, which will be in the focus of the further consideration. In the fast modulation regime one obtains good approximation by employing the plot (d1/2)2 versus (xrf2/ce2). In order to increase the accuracy of B1 the contribution of Bessel functions for known parameters x2 and xrf of applied modulation frequency should be calculated (Eq. (5)). Moreover, one can also simulate complete experimental AM-CW-EPR spectrum by employing Eq. (2) to evaluate B1 in which case two additional parameters (T1 and T2) are required. 2.2. Simulation of AM-CW-EPR with Bloch-Siegert-like shift The expression given in Eq. (2) can be used to simulate AM-CWEPR spectra of a homogeneous line at various modulation frequencies and also to examine dependence of these spectra on the
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microwave field strength. It should be emphasized that some of these experimental data can be found in the literature [12–14] but were not discussed in the perspective of B1 [13,14]. One of these examples can be taken from Ref. [12]. (Fig. 2 in Ref. [12].), which shows series of experimental AM-CW-EPR spectra with the very narrow linewidth, dBL 1.5 µT (42 kHz), of a single crystal of fluoranthenyl radical recorded at various modulation frequencies applying the same microwave power. For the present analyses the separation between sidebands in the spectra, due to absence of their digital values, was recycled from the picture and presented in Fig. 1. AM-CW-EPR spectra for fluoranthenyl radical cation were simulated by employing Eq. (2) and assuming that T1 = T2, where T2 is related to Lorentzian linewidth as 2T2 = (cedBL)1. The same series of modulation frequencies (0.2–1.5 MHz) as in Ref. [12] were considered and the results presented in Fig. 2. The parameters used in the simulation generally satisfy the condition for ‘‘strong and fast modulation” regime. It can be noted that the series of simulated AM-CW-EPR spectra qualitatively describe the similar shape and splitting of the experimental AM-CW-EPR spectra reported in the literature (Fig. 2 in Ref. [12]). The net effect of Dk on AM-CW-EPR lineshape and splitting can be seen for the calculation performed for the same xrf (Fig. 2). It should be emphasized that the simulated AM-CW-EPR spectra are very close to the rapid passage condition (xrf T2 > 1) and that a relative small change in ratio xrf/x2 and T1/T2 will significantly influence the AM-CW-EPR lineshape [13,14]. The calculated separations between the first sideband peaks at various modulation frequencies are also presented in Fig. 1. However, the small nonlinearity of experimental data can be noted while more accurate fit of data can be achieved when assuming two instead of one linear dependency. Therefore, in the present case the intercept of the straight line obtained in the region of small xrf values (calculated from the smallest four points in Fig. 1 according to Eq. (1) provides the estimation of B1 = 5.3 µT. This result deviates from the value of B1 = 6.8 µT which is obtained from the intercept of the straight line considering the rest of the higher frequencies of xrf presented in Fig. 1. The discrepancy in estimation B1 from Eq. (1) leads to an average value of B1av = 6.0 µT with the uncertainty (±13%). However, description which includes Bloch-Siegert-like shift from Eq. (2) increases accuracy (within error of 5%) of evaluated B1. This was obtained with B1 = 6.4 µT as the best fitted value for description separation between sideband peaks for all series of AM-CWEPR spectra given in (Fig. 2, in Ref. [12]). The above consideration
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Fig. 2. X-band EPR simulated AM-CW-EPR spectra for fluoranthenyl radical cation at various values of modulation frequencies (xrf/2p). Simulation was obtained by employing Eq. (2) to simulate spectra in Ref. [12]. Parameters for calculation: dBL = 1.5 µT, x2/ce = 2 µT and B1 = 6.4 µT is chosen as the best fitted parameter. The dotted spectrum corresponds to simulated AM-CW-EPR spectrum with the same parameters as spectrum at 0.2 MHz but with neglected D1 term.
clearly shows possibility of estimation of B1 by employing description of AM-CW-EPR spectrum, which includes Bloch-Siegert-like shift. The limitation of this method is related to the selection of the modulation frequencies, which can be only from the specific frequency region (xrf > x1) while the linewidth of the sample probes should satisfy the condition of xrf > 1/T2. 2.3. The second harmonic detection of MS spectrum An additional example from the literature will be also discussed. The AM-CW-EPR spectra of lithium phthalocyanine (LiPc) (Fig. 3A in Ref. [13]) exhibit clear indication that sideband splitting depend on the increasing microwave power from low (0.63 mW) to higher value (20.02 mW). Significant change of the spectra recorded above 20.02 mW can be noticed due to saturation effect. In addition, it can clearly be seen that the second sidebands show slightly smaller separation/splitting when the spectrum was recorded at 20.02 mW than at 0.63mW while using the same modulation frequency. Such an effect is possible due to Bloch-Siegertlike shift when the modulation frequency is larger than the expected frequency of B1(xrf/2p = 1 MHz > x1/2p 0.3 MHz). The same type of the effect has an impact on the separation of the first sidebands, d1, but due to an increasing of the line broadening it cannot be clearly detected and evaluated from AM-CW-EPR spectrum. In order to avoid this problem and to increase resolution of spectral sidebands the methodology of the second harmonic detection is the subject of here presented study employing for the same type of the sample [13,14] at the similar RF and microwave power conditions. 3. Measurement of microwave field strength from MS spectra 3.1. Experimental
Fig. 1. Plot of data, (d1/2)2 versus (xrf/2p)2, open triangles denote experimental data from Ref. [12] and ‘‘x” denote calculated data from Eq. (2). The dashed straight lines describe slightly deviation of experimental data from the linearity (Eq. (1)) at low (or at high) modulation frequencies.
For the measurements of B1 a polycrystalline sample of LiPc was synthesized by electrochemical oxidation of Li2Pc [18]. It serves as an S = 1/2 model system with nearly isotropic g = 2.0020 value and usually exhibits composite spectrum due to the broad (dpp 0.1 mT) and the narrow (dBpp 0.01 mT) component which differ within an order of magnitude with respect to the linewidth.
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The contribution of narrow component can be enhanced by selection of several (poly) crystals (0.05 0.05 0.3 mm3) from the powder sample of LiPc or simply by recording the powder sample with small modulation amplitude (5 µT). In such case the contribution of broad component is significantly reduced in comparison with the narrow component of composite spectrum. As is expected, due to the polycrystalline structure of the LiPc sample the narrow component that is in the focus of the present study exhibits an inhomogeneous lineshape described with the Gaussian-type rather than simple Lorentzian. MS spectra were measured on commercial X-band pulse EPR spectrometer (Bruker Elexsys E580) accompanied with additional homemade RF modulation coil, which was inserted in the microwave cavity (rectangular TE102). The RF was produced from ENDOR unit, which was integrated in the spectrometr, and RF was amplified by broad-band amplifier (EIN model 3100L). The inserted coil was oriented to produce RF modulation field parallel to the static magnetic field. LiPc sample was placed in the center of RF coil. The microwave cavity containing RF coil was tuned and critically coupled as in the standard CW EPR experiment. MS spectrum was recorded in accord with previously suggested method [19] using standard modulation frequency of xm/2p = 50 kHz and modulation amplitude of Bm = 1 µT along with the second harmonic detection. EPR spectrum is proportional to the second derivative of the zeroth-harmonic modulation spectrum as was shown and discussed earlier [19]. The approach based on the detection using second harmonic shows partial advantage but also partial disadvantage when compared with the AM-CW-EPR detection. The advantage of detection is due to the fact that the sideband peaks coincide with peaks of the zeroth-harmonic and while being narrower than the peaks of zeroth-harmonic, they increase the spectral resolution of sidebands. On the other side, disadvantage is related to the reduced sensitivity and an additional curvature of the base line. In order to evaluate B1 by employing Eq. (4) or Eq. (5) one needs the most precise information about the mutual position of the peaks and their corresponding intensities and hence for this type of the study the choice of the second harmonic detection is advantageous.
3.2. Experimental results and discussion Fig. 3 shows typical spectrum of induced sidebands for EPR absorption singlet of the LiPc sample detected at the second
Fig. 3. MS spectrum detected at the second harmonic (xm/2p = 50 kHz, Bm = 1 µT) of CW-EPR (continuous line) and numerically simulated (dashed line) by employing the second derivative of the first term of Eq. (2). Parameters: T1 = 0.45 µs, T2 = 0.45 µs, xrf/2p = 1 MHz, x2/2p = 0.4 MHz and x1/2p = 0.36 MHz.
harmonic. The spectrum was detected by applying the RF modulation field with the corresponding xrf/2p = 1 MHz and x2/2p 0.2 MHz frequencies. The two outer peaks visible on the spectrum represent first sidebands peaks with expected separation of d1 = 0.0714 mT 2 MHz while the peak in the center of the spectrum represents the central sideband. This MS spectrum can be simulated by taking the second derivative of the first term in Eq. (2). Due to the presence of Bloch-Siegert-like shift contribution it is expected that simulation of MS spectrum will better describe experimental spectrum than the earlier suggested relation [13,14] for AM-CW-EPR without such correction term. It can be noticed that simulated spectrum (Fig. 3) reproduce well the sideband peaks positions as well as intensities. There is a certain deviation in the lineshape observable in the vicinity of the base line, which is due to the assumed Lorentzian lineshape for the inhomogeneous LiPc spectral line. One can approximately deduce the value of z and obtained Brf by employing the ratio of intensities between the one of the first sideband peaks (I1) and the central sideband peak (I0) [19]. For small z values (z 1) approximation I1/I0 = J21(z)/J20(z) was used. It is important to note that for the estimation of reliable z values very low saturation (S 1) should be applied. This is a consequence of non-equal saturation of the central versus other sidebands at higher saturations [2]. Thus, estimation of z value for all the examined RF frequencies at the lowest possible saturation level could be also very useful for the deducing B1 as presented in Eqs. (4) and (5) and discussed above. In order to use modulation induced sidebands for measurement B1 the series of spectra as a function of xrf in the frequency interval from 1–2 MHz were recorded at the same microwave power (Fig. 4). They reflect how d1 increases with increasing xrf as was expected from the first term in Eq. (2). This phenomenon can be presented in the scaled plot of (d1/2)2(1 z2/4)2 versus (xrf/ce)2(1 + D21/xrf2)(1 z2/4)2 in which the intercept of the line at abscissa indicates measurable effect produced by B21 value. Experimental data referring to the with the same parameters as presented in Fig. 4 except for the variation of the microwave power are shown in Fig. 5. Here the comparison between the same data with and without corrections is presented in the same plot. Evaluated B1 values derived from Eq. (1) and improved approximation given in Eq. (3) are collected in Table 1. It can be easily noted that the obtained data for the corresponding values of microwave power describe mutually nearly parallel straight lines with
Fig. 4. Typical MS spectra of LiPc detected at the second harmonic (xm/2p = 50 kHz, Bm = 1 µT) of CW-EPR for different values of the modulation frequency, xrf/2p at constant P = 2 mW and x2. Distance between the first sideband peaks is denoted with d1.
B. Rakvin et al. / Journal of Magnetic Resonance 287 (2018) 123–127
127
to the first order in x1/xrf. In the present work this method was extended for the larger values of B1 where condition of x1/xrf 1 is changed towards larger values (x1/xrf < 1) and higher order perturbation terms which were taken into account. It is shown that by introducing Bloch-Siegert-like shift in Eq. (2) it is possible simulated AM-CW-EPR spectra which can be further used to deduce B1. In order to increase sensitivity of this method in the region of higher intensity of B1 where the saturation effects decrease the resolution of AM-CW-EPR spectra the additional second harmonic of CW EPR can be successfully employed. MS spectrum appears to be detected with well resolved sideband peaks and can be used to evaluate B1 in the presence of saturation effects. Acknowledgment
Fig. 5. Measured data for d1 presented in the scaled plot (filled symbols) at different microwave power (20 mW triangles and 201 mW squares). Data are fitted with dashed straight line to obtained intercept (B21). The open symbols with corresponding dotted straight lines represent the same data without included corrections (Eq. (1)).
Table 1 Evaluated B1 values from approximation (Eq. (1)) and improved approximation (Eq. (5)) at various microwave power. The employed probe is the narrow line (dBpp 0.01 mT) of LiPc sample. P (mW)
B1 (mT) calculated from Eq. (1)
B1 (mT) calculated from Eq. (5)
20 201
0.0115 ± 0.0022 0.0370 ± 0.0012
0.0137 ± 0.0020 0.0407 ± 0.0021
different intercepts (B21) on the abscissa. The effect of correction due to second order contribution in the present case can be clearly seen only at the highest employed microwave power as a difference between the dashed and continuous straight lines (Fig. 5). This result also shows that in the limit of high xrf values (xrf x1) the approach with here presented corrections converges to the approximate solution presented by Eq. (1). In addition, the impact of the linewidth on the measured accuracy of B1 is clearly seen. As expected the accuracy of estimated B1 decreases when the linewidth of the sample probe is larger or comparable with the B1. Here presented results are complementary with the saturation behavior of LiPc sideband spectrum detected by AM-CW-EPR given previously in Fig. 3A [13]. There is no detectable change between sidebands separations in the low power regime (<20 mW). The effect starts to be visible on the spectrum acquired at 20 mW. However, the later spectrum shows significant broadening in the region between the first sideband and it is not convenient for the accurate measurement of d1. By employing the second harmonic detection the resolution of MS spectrum increase and makes possible measurement of d1 with higher accuracy at 20 mW and at other higher microwave powers. 4. Conclusions MS in the EPR spectrum was used for measuring B1 in CW-EPR experiment under the condition of strong and fast modulation regime (x2 > x1 xrf). In the earlier suggested method only the case of x1 xrf was considered and the results can be reduced
This work was supported in part by the Croatian Science Foundation under the project number 1108. References [1] O. Haworth, R.E. Richards, The use of modulation in magnetic resonance, Prog. Nucl. Magn. Reson. Spectrosc. 1 (1966) 1–14. [2] M. Kalin, I. Gromov, A. Schweiger, The continuous wave electron paramagnetic resonance experiment revisited, J. Magn. Reson. 160 (2003) 166–182. [3] M. Kälin, M. Fedin, I. Gromov, A. Schweiger, Multiple-Photon Transitions in EPR Spectroscopy, in: J. Dolinšek, M. Vilfan, S. Zˇumer (Eds.), Novel NMR and EPR Techniques, Springer Berlin Heidelberg, Berlin, Heidelberg, 2006, pp. 143–183. [4] E.L. Ginzton, Microwave Measurements, McGraw-Hill, New York, 1957. [5] J.H. Freed, D.S. Leniart, J.S. Hyde, Theory of saturation and double resonance effects in ESR Spectra. III. rf coherence and line shapes, J. Chem. Phys. 47 (1967) 2762–2773. [6] D.P. Dalal, S.S. Eaton, G.R. Eaton, The effects of lossy solvents on quantitative EPR studies, J. Magn. Reson. 44 (1981) (1969) 415–428. [7] K. Balasubramanian, A.H. Beth, B.H. Robinson, L.R. Dalton, S.D. Venkataramu, J. H. Park, Sensitivity of V2’ saturation transfer electron paramagnetic resonance signals to anisotropic rotational diffusion with [15N]nitroxide spin-labels. Effects of noncoincident magnetic and diffusion tensor principal axes, J. Phys. Chem. 87 (1983) 359–367. [8] M.A. Hemminga, F.A.M. Leermakers, P.A. de Jager, Quantitative measurement of B1 in ESR and saturation-transfer ESR spectroscopy, J. Magn. Reson. 59 (1984) (1969) 137–140. [9] A.I. Vistnes, L.R. Dalton, Experimental methods to determine the microwave field strength in electron spin resonance, J. Magn. Reson. 54 (1983) (1969) 78– 88. [10] C. Mailer, H. Thomann, B.H. Robinson, L.R. Dalton, Crossed TM110 bimodal cavity for measurement of dispersion electron paramagnetic resonance and saturation transfer electron paramagnetic resonance signals for biological materials, Rev. Sci. Instrum. 51 (1980) 1714–1721. [11] W.A. Anderson, Nuclear magnetic resonance spectra of some hydrocarbons, Phys. Rev. 102 (1956) 151–167. [12] M. Peric, B. Rakvin, A. Dulcic, Measurement of microwave field-strength in electron-spin-resonance by a pulsed modulation technique, J. Magn. Reson. 65 (1985) 215–221. [13] M. Fedin, I. Gromov, A. Schweiger, Absorption line CW EPR using an amplitude modulated longitudinal field, J. Magn. Reson. 171 (2004) 80–89. [14] M. Fedin, I. Gromov, A. Schweiger, Sensitivity optimization in amplitudemodulated CW-EPR experiment, J. Magn. Reson. 182 (2006) 293–297. [15] A.P. Saiko, G.G. Fedoruk, S.A. Markevich, Multiphoton transitions in a spin system driven by strong bichromatic field, J. Exp. Theor. Phys. 105 (2007) 893– 899. [16] A.P. Saiko, R. Fedaruk, S.A. Markevich, Multi-photon transitions and Rabi resonance in continuous wave EPR, J. Magn. Reson. 259 (2015) 47–55. [17] A.P. Saiko, G.G. Fedoruk, Effect of the Bloch-Siegert shift on the frequency responses of Rabi oscillations in the case of nutation resonance, JETP Lett. 87 (2008) 128–132. [18] G. Ilangovan, J.L. Zweier, P. Kuppusamy, Electrochemical preparation and EPR studies of lithium phthalocyanine: evaluation of the nucleation and growth mechanism and evidence for potential-dependent phase formation, J. Phys. Chem. B 104 (2000) 4047–4059. [19] I. Miyagawa, Y. Hayashi, Y. Kotake, Modulation-induced sidebands in magnetic resonance, J. Magn. Reson. 25 (1977) (1969) 183–195.
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Enhanced accuracy of the microwave field strength measurement in a CW-EPR by pulsed modulation technique B. Rakvin ⇑, D. Caric´, M. Kveder Ruder Boškovic´ Institute, Division of Physical Chemistry, Bijenicˇka 54, Zagreb, Croatia
a r t i c l e
i n f o
Article history: Received 11 December 2017 Revised 9 January 2018 Accepted 10 January 2018 Available online 11 January 2018 Keywords: EPR spectroscopy Microwave field strength Modulation spectrum Amplitude modulation EPR Bloch-Siegert shift
a b s t r a c t The microwave magnetic field strength, B1, in the cavity of a conventional continuous wave electron paramagnetic resonance, CW-EPR, spectrometer was measured by employing modulation sidebands, MS, in the EPR spectrum. MS spectrum in CW-EPR is produced by applying the modulation frequency, xrf, which exceeds the linewidth, dB, given in frequency units. An amplitude-modulated CW-EPR, AM-CW-EPR, was selected as detection method. Theoretical description of AM-CW-EPR spectrum was modified by adding Bloch-Siegert-like shift obtained by taking into account the cumulative effect of the non-resonant interactions between the driving fields and the spin system. This approach enables to enhance the precision of B1 measurement. In order to increase the sensitivity of the method when saturation effects, due to higher intensity of B1, decrease the resolution of AM-CW-EPR spectrum, detection at the second harmonic of CW-EPR has been employed. Ó 2018 Elsevier Inc. All rights reserved.
1. Introduction The microwave magnetic field strength, B1, as well as closely related Rabi frequency (x1 = ceB1) of a spin system in the microwave field, MW, are essential parameters for all quantitative measurements in EPR spectroscopy. In recent years studies of spin system containing very long relaxation times (transverse, T2 and longitudinal, T1) are of interest due to their potential application in quantum information technologies. One of the most often addressed model cases in these studies is the resonant interaction between the bichromatic field (transverse MW field and longitudinal radiofrequency (RF) field) and two level spin system (spin qubit) experiencing the Zeeman magnetic field. Early theoretical study of modulation effect in EPR spectroscopy was based on semi-classical approach by applying modified Bloch equations [1]. However, recently the description was revised and modulation effects in the continuous wave EPR, CW-EPR, spectroscopy explained by introducing multi-photon transitions [2,3]. By using quantized radiation fields it was shown [2,3] that modulation sidebands, MS, appearing in the EPR spectrum when the modulation frequency, xrf, used in CW-EPR exceeds the linewidth, dB, can be described by the multi-photon transitions. In these transitions one MW r+ photon is absorbed from the MW radiation field and
⇑ Corresponding author. E-mail address: [email protected] (B. Rakvin). https://doi.org/10.1016/j.jmr.2018.01.006 1090-7807/Ó 2018 Elsevier Inc. All rights reserved.
an arbitrary number k of RF p photons are absorbed from or emitted to the modulation RF field (r+ + k p). In the present consideration among the various different methods [4–11] for estimation of B1, the experimental method with pulsed modulation detection proposed several decades ago [12] will be discussed and revised within recently suggested multiple photon description of CW-EPR spectra. The method is based on the measurements of splitting between the first sideband signals [10,11], d1, under the small modulation index z = 2x2/xrf (2x2 corresponds to the amplitude of RF field), (z 1) [12]
2 x2rf d1 ¼ 2 B21 2 ce
ð1Þ
One can measure d1 for several values of xrf, and from the intercept on the abscissa of the plot (d1/2)2 versus xrf2 the value of B1 can be obtained in accord with Eq. (1). This approach shows several advantages in comparison with other methods [4,5,8,9]. As was shown earlier the method [12] requires no additional calibration procedure and yields B1 values directly in a wide range of microwave power. Moreover, the pulsed modulation technique can be performed by employing standard audiofrequency lock-in amplifier for detection of the dc signal component rather than the first harmonic component of a sideband spectrum. It could be noted that detection of the dc signal of sidebands spectrum is possible with low frequency pulse modulation (within range of 0–100 kHz as expected for lock-in amplifier included in a standard CW-EPR spectrometer) in
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comparison with the first harmonic detection [10,11] where significantly large xrf is usually required (xrf 1 MHz) for producing sideband’s splitting and lock-in detection at the same frequency [2]. However, there is some limitation in this method due to approximation in theoretical description, which is based on the semi-classical rotation frame model [2]. The sideband’s splitting taken for close sideband’s resonance (where condition (x1/xrf 1) is not preserved and transforms towards the condition (x1/xrf < 1)) deviate from the straight line in the plot d21 versus xrf2 [12]. Therefore, these experimental points were avoided in the process of evaluation of the B1. 2. Effect of Bloch-Siegert-like shift on MS spectrum 2.1. Amplitude-modulated CW-EPR with Bloch-Siegert-like shift The experimental method of pulsed modulation detection employed earlier generally coincides with more recently suggested method of amplitude-modulated CW-EPR, AM-CW-EPR [13,14]. Thus, in order to simulate experimental modulation spectra detected previously [12] and to evaluate more accurately the corresponding B1, the theoretical description of the AM-CWEPR spectrum for an individual homogeneous line is required. Presently suggested theoretical description [13,14] includes B1 exclusively in the saturation term, while the splitting between sidebands is lacking the B1 dependence and it is proportional only to the multiple of modulation frequency (kxrf). Therefore, here presented study improves this description by introducing Bloch-Siegert-like shift, Dk, of the energy levels obtained from the expanded Hamiltonian for resonant multiple photon transitions [2,3,15,16]. Following recent theoretical description of spin dynamics which can be treated with frequency parameters in the range of CW-EPR spectroscopy [16], it is convenient to introduce two different regimes of the spin dynamics such as: ‘‘weak modulation near the Rabi resonance” and ‘‘strong and fast modulation” with the corresponding values of the applied RF modulation (x2 x1 xrf) and (x2 > x1 xrf), respectively. For each of these regimes the corresponding Dk was suggested and calculated. In accord with the selection of RF for the ‘‘strong and fast modulation” regime Dk was calculated [15,16] up to the second order contribution in small parameter x1/xrf. By employing these results and an additional term of Dk for the evaluation a homogenous EPR line [16] the new description of AM-CW-EPR spectrum [13,14] is here suggested 1 X
P¼
"
x1 T 2 J2k ðzÞ 2
k¼1
Dk ¼
1 þ x21 T 1 T 2 J 2k ðzÞ þ ðXs kxrf þ Dk Þ T 22 # x1 T 2 J2k ð0Þ 2
1 þ x21 T 1 T 2 J 2k ð0Þ þ ðXs kxrf þ Dk Þ T 22
ð2Þ
2 X 2 x21 x1 J l ðzÞ þ 4xmw 2xrf l–k ðk lÞ
where z = 2x2/xrf, Jk(z) is the Bessel function of the first kind and order k, Xs = xs xmw and xs = ceB0. For weak MW fields, as commonly used in CW-EPR the first term in Dk is small and can be neglected. It can be shown that Dk = Dk and there is no shift for D0 = 0 (k = 0, centerband) while the largest contribution to the splitting is expected between the first sidebands in comparison with the splitting between the second and higher order of sidebands. From the Eq. (2) one can easily evaluate the new corrected splitting between sidebands peaks as:
ce dk 2
¼ kxrf Dk
ð3Þ
In order to compare Eqs. (1) and (3), Eq. (3) can be transformed in the form of Eq. (1)
!2 2 X 2 2 X J 2 ðzÞ x2rf d1 1 x1 J l ðzÞ 2 l ¼ 2 B1 þ ð1 lÞ c2e 2xrf ð1 lÞ 2 ce l–1 l–1
ð4Þ
The obtained result approaches the result derived from the rotation frame model given in Eq. (1) when assuming conditions for strong and fast modulation (z 1, and x1/xrf 1). Under these conditions and by taking the leading terms in the Bessel sum approaching unit value the Eq. (4) is further simplified
2 2 x2rf d1 z2 1 2 2 4 ce
1þ
D21
x2rf
!
1
z2 4
2 B21
ð5Þ
The obtained relation increases accuracy of a description of the intercept on the abscissa of the plot (d1/2)2 versus xrf2 in comparison with the earlier description given by Eq. (1). The effect of these corrections can be easily followed by presenting the data using scaled plot (d1/2)2(1 z2/4)2 versus (xrf/ce)2(1 z2/4)2 (1 + D21/xrf2). This approach includes the change of z and D1 parameters as is expected for measurements at different xrf values. Additionally, the similar analyses can be provided for the second spin dynamics regime ‘‘weak modulation near the Rabi resonance” and the corrected separation between the sidebands peaks is expected as follows [16]
2 x2rf Dð1Þ2 d1 ¼ 2 1 B21 2 ce xrf "
Dð1Þ ¼
x22 sin2 h X 1 2 ðJ ðaÞ þ J n ðaÞJ n2 ðaÞÞ 2xrf n n n–0 # 1 ðJ 2n ðaÞ þ J n ðaÞJ nþ2 ðaÞÞ nþ2 n–2
þ
X
ð6Þ
where a = (2x2/xrf) cos h, sin h = x1/(x21 + Xs2)1/2 and cos h = Xs/ (x21 + Xs2)1/2. In this dynamics regime D(1) is calculated up to the second order contribution by assuming (x2/xrf) sin h 1 as a small parameter. The advantage of this theoretical result is that one can describe the splitting between sidebands when RF modulation is in the vicinity of Rabi resonance (x1 xrf) [16,17]. It is expected that d1 exhibits minimum at resonance condition (x1 = xrf) and amplitudes of low-field and high-field sideband are not equal as was demonstrated in the time resolved EPR experiment [17]. Thus, the choice of the appropriate relation to express measured data in terms of d21 versus xrf2 at constant microwave power depends on the characteristic relaxation parameters T1 and T2 of spin system and on the applied regime of spin dynamics (x2, xrf and x1). One expects that for more reliable measurements of d1 it is convenient to use larger values of, d1 2xrf than the smaller values of d1 near the low limit, d1 dB. This requires measurements in the fast motional regime, which will be in the focus of the further consideration. In the fast modulation regime one obtains good approximation by employing the plot (d1/2)2 versus (xrf2/ce2). In order to increase the accuracy of B1 the contribution of Bessel functions for known parameters x2 and xrf of applied modulation frequency should be calculated (Eq. (5)). Moreover, one can also simulate complete experimental AM-CW-EPR spectrum by employing Eq. (2) to evaluate B1 in which case two additional parameters (T1 and T2) are required. 2.2. Simulation of AM-CW-EPR with Bloch-Siegert-like shift The expression given in Eq. (2) can be used to simulate AM-CWEPR spectra of a homogeneous line at various modulation frequencies and also to examine dependence of these spectra on the
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microwave field strength. It should be emphasized that some of these experimental data can be found in the literature [12–14] but were not discussed in the perspective of B1 [13,14]. One of these examples can be taken from Ref. [12]. (Fig. 2 in Ref. [12].), which shows series of experimental AM-CW-EPR spectra with the very narrow linewidth, dBL 1.5 µT (42 kHz), of a single crystal of fluoranthenyl radical recorded at various modulation frequencies applying the same microwave power. For the present analyses the separation between sidebands in the spectra, due to absence of their digital values, was recycled from the picture and presented in Fig. 1. AM-CW-EPR spectra for fluoranthenyl radical cation were simulated by employing Eq. (2) and assuming that T1 = T2, where T2 is related to Lorentzian linewidth as 2T2 = (cedBL)1. The same series of modulation frequencies (0.2–1.5 MHz) as in Ref. [12] were considered and the results presented in Fig. 2. The parameters used in the simulation generally satisfy the condition for ‘‘strong and fast modulation” regime. It can be noted that the series of simulated AM-CW-EPR spectra qualitatively describe the similar shape and splitting of the experimental AM-CW-EPR spectra reported in the literature (Fig. 2 in Ref. [12]). The net effect of Dk on AM-CW-EPR lineshape and splitting can be seen for the calculation performed for the same xrf (Fig. 2). It should be emphasized that the simulated AM-CW-EPR spectra are very close to the rapid passage condition (xrf T2 > 1) and that a relative small change in ratio xrf/x2 and T1/T2 will significantly influence the AM-CW-EPR lineshape [13,14]. The calculated separations between the first sideband peaks at various modulation frequencies are also presented in Fig. 1. However, the small nonlinearity of experimental data can be noted while more accurate fit of data can be achieved when assuming two instead of one linear dependency. Therefore, in the present case the intercept of the straight line obtained in the region of small xrf values (calculated from the smallest four points in Fig. 1 according to Eq. (1) provides the estimation of B1 = 5.3 µT. This result deviates from the value of B1 = 6.8 µT which is obtained from the intercept of the straight line considering the rest of the higher frequencies of xrf presented in Fig. 1. The discrepancy in estimation B1 from Eq. (1) leads to an average value of B1av = 6.0 µT with the uncertainty (±13%). However, description which includes Bloch-Siegert-like shift from Eq. (2) increases accuracy (within error of 5%) of evaluated B1. This was obtained with B1 = 6.4 µT as the best fitted value for description separation between sideband peaks for all series of AM-CWEPR spectra given in (Fig. 2, in Ref. [12]). The above consideration
125
Fig. 2. X-band EPR simulated AM-CW-EPR spectra for fluoranthenyl radical cation at various values of modulation frequencies (xrf/2p). Simulation was obtained by employing Eq. (2) to simulate spectra in Ref. [12]. Parameters for calculation: dBL = 1.5 µT, x2/ce = 2 µT and B1 = 6.4 µT is chosen as the best fitted parameter. The dotted spectrum corresponds to simulated AM-CW-EPR spectrum with the same parameters as spectrum at 0.2 MHz but with neglected D1 term.
clearly shows possibility of estimation of B1 by employing description of AM-CW-EPR spectrum, which includes Bloch-Siegert-like shift. The limitation of this method is related to the selection of the modulation frequencies, which can be only from the specific frequency region (xrf > x1) while the linewidth of the sample probes should satisfy the condition of xrf > 1/T2. 2.3. The second harmonic detection of MS spectrum An additional example from the literature will be also discussed. The AM-CW-EPR spectra of lithium phthalocyanine (LiPc) (Fig. 3A in Ref. [13]) exhibit clear indication that sideband splitting depend on the increasing microwave power from low (0.63 mW) to higher value (20.02 mW). Significant change of the spectra recorded above 20.02 mW can be noticed due to saturation effect. In addition, it can clearly be seen that the second sidebands show slightly smaller separation/splitting when the spectrum was recorded at 20.02 mW than at 0.63mW while using the same modulation frequency. Such an effect is possible due to Bloch-Siegertlike shift when the modulation frequency is larger than the expected frequency of B1(xrf/2p = 1 MHz > x1/2p 0.3 MHz). The same type of the effect has an impact on the separation of the first sidebands, d1, but due to an increasing of the line broadening it cannot be clearly detected and evaluated from AM-CW-EPR spectrum. In order to avoid this problem and to increase resolution of spectral sidebands the methodology of the second harmonic detection is the subject of here presented study employing for the same type of the sample [13,14] at the similar RF and microwave power conditions. 3. Measurement of microwave field strength from MS spectra 3.1. Experimental
Fig. 1. Plot of data, (d1/2)2 versus (xrf/2p)2, open triangles denote experimental data from Ref. [12] and ‘‘x” denote calculated data from Eq. (2). The dashed straight lines describe slightly deviation of experimental data from the linearity (Eq. (1)) at low (or at high) modulation frequencies.
For the measurements of B1 a polycrystalline sample of LiPc was synthesized by electrochemical oxidation of Li2Pc [18]. It serves as an S = 1/2 model system with nearly isotropic g = 2.0020 value and usually exhibits composite spectrum due to the broad (dpp 0.1 mT) and the narrow (dBpp 0.01 mT) component which differ within an order of magnitude with respect to the linewidth.
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The contribution of narrow component can be enhanced by selection of several (poly) crystals (0.05 0.05 0.3 mm3) from the powder sample of LiPc or simply by recording the powder sample with small modulation amplitude (5 µT). In such case the contribution of broad component is significantly reduced in comparison with the narrow component of composite spectrum. As is expected, due to the polycrystalline structure of the LiPc sample the narrow component that is in the focus of the present study exhibits an inhomogeneous lineshape described with the Gaussian-type rather than simple Lorentzian. MS spectra were measured on commercial X-band pulse EPR spectrometer (Bruker Elexsys E580) accompanied with additional homemade RF modulation coil, which was inserted in the microwave cavity (rectangular TE102). The RF was produced from ENDOR unit, which was integrated in the spectrometr, and RF was amplified by broad-band amplifier (EIN model 3100L). The inserted coil was oriented to produce RF modulation field parallel to the static magnetic field. LiPc sample was placed in the center of RF coil. The microwave cavity containing RF coil was tuned and critically coupled as in the standard CW EPR experiment. MS spectrum was recorded in accord with previously suggested method [19] using standard modulation frequency of xm/2p = 50 kHz and modulation amplitude of Bm = 1 µT along with the second harmonic detection. EPR spectrum is proportional to the second derivative of the zeroth-harmonic modulation spectrum as was shown and discussed earlier [19]. The approach based on the detection using second harmonic shows partial advantage but also partial disadvantage when compared with the AM-CW-EPR detection. The advantage of detection is due to the fact that the sideband peaks coincide with peaks of the zeroth-harmonic and while being narrower than the peaks of zeroth-harmonic, they increase the spectral resolution of sidebands. On the other side, disadvantage is related to the reduced sensitivity and an additional curvature of the base line. In order to evaluate B1 by employing Eq. (4) or Eq. (5) one needs the most precise information about the mutual position of the peaks and their corresponding intensities and hence for this type of the study the choice of the second harmonic detection is advantageous.
3.2. Experimental results and discussion Fig. 3 shows typical spectrum of induced sidebands for EPR absorption singlet of the LiPc sample detected at the second
Fig. 3. MS spectrum detected at the second harmonic (xm/2p = 50 kHz, Bm = 1 µT) of CW-EPR (continuous line) and numerically simulated (dashed line) by employing the second derivative of the first term of Eq. (2). Parameters: T1 = 0.45 µs, T2 = 0.45 µs, xrf/2p = 1 MHz, x2/2p = 0.4 MHz and x1/2p = 0.36 MHz.
harmonic. The spectrum was detected by applying the RF modulation field with the corresponding xrf/2p = 1 MHz and x2/2p 0.2 MHz frequencies. The two outer peaks visible on the spectrum represent first sidebands peaks with expected separation of d1 = 0.0714 mT 2 MHz while the peak in the center of the spectrum represents the central sideband. This MS spectrum can be simulated by taking the second derivative of the first term in Eq. (2). Due to the presence of Bloch-Siegert-like shift contribution it is expected that simulation of MS spectrum will better describe experimental spectrum than the earlier suggested relation [13,14] for AM-CW-EPR without such correction term. It can be noticed that simulated spectrum (Fig. 3) reproduce well the sideband peaks positions as well as intensities. There is a certain deviation in the lineshape observable in the vicinity of the base line, which is due to the assumed Lorentzian lineshape for the inhomogeneous LiPc spectral line. One can approximately deduce the value of z and obtained Brf by employing the ratio of intensities between the one of the first sideband peaks (I1) and the central sideband peak (I0) [19]. For small z values (z 1) approximation I1/I0 = J21(z)/J20(z) was used. It is important to note that for the estimation of reliable z values very low saturation (S 1) should be applied. This is a consequence of non-equal saturation of the central versus other sidebands at higher saturations [2]. Thus, estimation of z value for all the examined RF frequencies at the lowest possible saturation level could be also very useful for the deducing B1 as presented in Eqs. (4) and (5) and discussed above. In order to use modulation induced sidebands for measurement B1 the series of spectra as a function of xrf in the frequency interval from 1–2 MHz were recorded at the same microwave power (Fig. 4). They reflect how d1 increases with increasing xrf as was expected from the first term in Eq. (2). This phenomenon can be presented in the scaled plot of (d1/2)2(1 z2/4)2 versus (xrf/ce)2(1 + D21/xrf2)(1 z2/4)2 in which the intercept of the line at abscissa indicates measurable effect produced by B21 value. Experimental data referring to the with the same parameters as presented in Fig. 4 except for the variation of the microwave power are shown in Fig. 5. Here the comparison between the same data with and without corrections is presented in the same plot. Evaluated B1 values derived from Eq. (1) and improved approximation given in Eq. (3) are collected in Table 1. It can be easily noted that the obtained data for the corresponding values of microwave power describe mutually nearly parallel straight lines with
Fig. 4. Typical MS spectra of LiPc detected at the second harmonic (xm/2p = 50 kHz, Bm = 1 µT) of CW-EPR for different values of the modulation frequency, xrf/2p at constant P = 2 mW and x2. Distance between the first sideband peaks is denoted with d1.
B. Rakvin et al. / Journal of Magnetic Resonance 287 (2018) 123–127
127
to the first order in x1/xrf. In the present work this method was extended for the larger values of B1 where condition of x1/xrf 1 is changed towards larger values (x1/xrf < 1) and higher order perturbation terms which were taken into account. It is shown that by introducing Bloch-Siegert-like shift in Eq. (2) it is possible simulated AM-CW-EPR spectra which can be further used to deduce B1. In order to increase sensitivity of this method in the region of higher intensity of B1 where the saturation effects decrease the resolution of AM-CW-EPR spectra the additional second harmonic of CW EPR can be successfully employed. MS spectrum appears to be detected with well resolved sideband peaks and can be used to evaluate B1 in the presence of saturation effects. Acknowledgment
Fig. 5. Measured data for d1 presented in the scaled plot (filled symbols) at different microwave power (20 mW triangles and 201 mW squares). Data are fitted with dashed straight line to obtained intercept (B21). The open symbols with corresponding dotted straight lines represent the same data without included corrections (Eq. (1)).
Table 1 Evaluated B1 values from approximation (Eq. (1)) and improved approximation (Eq. (5)) at various microwave power. The employed probe is the narrow line (dBpp 0.01 mT) of LiPc sample. P (mW)
B1 (mT) calculated from Eq. (1)
B1 (mT) calculated from Eq. (5)
20 201
0.0115 ± 0.0022 0.0370 ± 0.0012
0.0137 ± 0.0020 0.0407 ± 0.0021
different intercepts (B21) on the abscissa. The effect of correction due to second order contribution in the present case can be clearly seen only at the highest employed microwave power as a difference between the dashed and continuous straight lines (Fig. 5). This result also shows that in the limit of high xrf values (xrf x1) the approach with here presented corrections converges to the approximate solution presented by Eq. (1). In addition, the impact of the linewidth on the measured accuracy of B1 is clearly seen. As expected the accuracy of estimated B1 decreases when the linewidth of the sample probe is larger or comparable with the B1. Here presented results are complementary with the saturation behavior of LiPc sideband spectrum detected by AM-CW-EPR given previously in Fig. 3A [13]. There is no detectable change between sidebands separations in the low power regime (<20 mW). The effect starts to be visible on the spectrum acquired at 20 mW. However, the later spectrum shows significant broadening in the region between the first sideband and it is not convenient for the accurate measurement of d1. By employing the second harmonic detection the resolution of MS spectrum increase and makes possible measurement of d1 with higher accuracy at 20 mW and at other higher microwave powers. 4. Conclusions MS in the EPR spectrum was used for measuring B1 in CW-EPR experiment under the condition of strong and fast modulation regime (x2 > x1 xrf). In the earlier suggested method only the case of x1 xrf was considered and the results can be reduced
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