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Unusual behaviour of a very narrow electron spin resonance signal
Journal of Magnetic Resonance 227 (2013) 46–50
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Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Unusual behaviour of a very narrow electron spin resonance signal François Beuneu Laboratoire des Solides Irradiés, CNRS-CEA, École Polytechnique, F-91128 Palaiseau, France
a r t i c l e
i n f o
Article history: Received 10 October 2012 Revised 7 November 2012 Available online 23 November 2012 Keywords: A. Lithium metal colloids B. Electron spin resonance
a b s t r a c t Unusually narrow electron spin resonance lines can give birth to spectacular and interesting phenomena. We use here very pure lithium metal colloids created in electron-irradiated LiF, giving rise to a very narrow Li metal line. In these samples, three interesting phenomena are discussed and interpreted: signal saturation in particularly good conditions; oscillations appearing on strongly overmodulated lines; signal bistability, when the signal shape seems to depend on the sample magnetic history. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The present paper describes some interesting phenomena observed in electron spin resonance (ESR) experiments on samples giving very narrow resonance lines, i.e. linewidths lower than 0.1G = 10 lT. Among materials giving birth to such narrow lines, we chose to work with Li metal small clusters or colloids embedded in an insulating matrix, which were obtained by electron-irradiation of single crystals of lithium fluoride LiF. In the past, we made a detailed study of these colloids with ESR [1]. In the past, the creation of lithium precipitates in LiF by irradiation was initiated by Lambert et al. [2], who used thermal neutron irradiations. Such irradiations were also done by Kaplan and Bray [3] and by Stesmans and Wu [4]. Electron irradiation was used by L’vov et al. [5]. Even with pure electronic excitations, as with c-ray irradiation, Van den Bosch [6] reported the observation of Li metal in LiF, but unfortunately did not do ESR experiments. Under room-temperature irradiation, fluorine vacancies are created and group themselves into anion-vacancies clusters, which therefore contain only Li atoms. Depending on the electron flux and fluence, these clusters can grow and reach sizes of several hundreds of nm, and even in the lm range. These very pure lithium metal precipitates are ideal candidates for giving extremely narrow ESR lines coming from conduction electrons of the metal. Conduction electron spin resonance (CESR) was extensively studied in the past and is generally a phenomenon difficult to detect, and in most cases only near liquid helium temperature. We gave a review of CESR properties in metals in two papers [7,8]: Li is quite peculiar because spin–orbit interaction is very small in this light metal. This leads to a narrow CESR line, without temperature variation of the linewidth, a unique behaviour. We recall that the signal intensity is independent of temperature, like for E-mail address: [email protected] 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.11.011
every metal, because it is proportional to the paramagnetic susceptibility which is of Pauli-type for conduction electrons. The spin–orbit weakness is also responsible for the fact that the g-factor of Li conduction electrons is extremely close to the freeelectron value, g 2.0023. Indeed, VanderVen [9] made precise measurements of the Li g-factor and stated that ‘‘the shift of the conduction–electron g value relative to that of the free electron is dg = (6.1 ± 0.2) 106.’’ A simple application of Li-colloids samples in ESR is to use them as g-factor standards, because their g is precisely known and their narrow width enables a precise determination of the resonance position. After describing the experimental conditions (Section 2) and particularly the way samples are produced, the paper describes three interesting phenomena in which the narrow ESR linewidth is crucial: a precise study of line saturation (Section 3) with application to rf magnetic field measurement in the ESR cavity; under extreme overmodulation conditions, oscillations are observed on the spectra (Section 4) and an interpretation is given; in the case of large samples, spectra become different if magnetic field is increased or decreased (Section 5): this bistability is studied and interpreted as a purely instrumental effect. 2. Experimental Specimens of LiF of dimensions 3–4 mm2, 500–1000 lm thick, were prepared by cleaving LiF crystals along (1 0 0). They were then uniformly irradiated by 2.5-MeV electrons from our Van de Graaff accelerator. During electron bombardment, samples were fixed on a water-cooled copper target with a 10-lm-thick copper foil. The irradiation temperature, monitored by a thermocouple, was about 50 °C. The introduced Li colloids were then detected and characterised by CESR, with a Bruker ER 200D or EMX X-band spectrometer, in a rectangular double cavity (TE104 mode) with a quality factor near 3000.
F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
In Ref. [1], we described two strategies to get extremely narrow Li ESR lines. The first one consists in using strong electron flux and fluence (for instance 3.6 1019 e/cm2 = 5.8 C/cm2 at 40 lA/cm2): the obtained lorentzian line is directly very intense and narrow (610 lT). The second uses lower irradiation conditions, such as 1.6 1019 e/cm2 = 2.6 C/cm2 at 6 lA/cm2. The line is then quite broad, but after an annealing near 700 °C, typical linewidth values are very small, about 3–4 lT (30–40 mG). Three samples are used in the present study. The first one, used in Section 3, was submitted to a fluence of 2.33 C/cm2 and a beam current of 16 lA/cm2. The second one, used in Section 4, received 1.17 C/cm2 under 15 lA/cm2. These two samples were of the second type described above: they gave broad CESR lines (typically 0.1 mT) after irradiation and were annealed at high temperature (700 °C); after annealing, they fell into small pieces, with a typical volume about 1 mm3, and one piece for each sample was used for the experiments described here. The third sample, used in Section 5, was irradiated at 28 lA/cm2 and 1.81 C/cm2. It was bigger in size, with a weight of 2.5 mg, and gave a narrow line (9 lT) without annealing. Samples of the same batch than the two first samples were measured at variable temperature, from 4 K to 300 K. Their width and intensity did not vary with temperature. For all samples used here, the lineshape was symmetrical and generally lorentzian. The dysonian lineshape very often observed in CESR is due to an incomplete microwave penetration into the sample because of skin effect. In our case, at room temperature, the skin depth is larger than the Li particle size, so that no dysonian shape is observed.
3. Saturation A saturation ESR experiment was performed at room temperature on the first irradiated LiF sample. The maximum source power was 200 mW, and the microwave attenuator was varied from 0 to 50 dB. The modulation was set at 100 kHz frequency and 2 lT peak-to-peak amplitude. The ESR line was adjusted numerically with a lorentzian shape by a least-squares program and the peak-to-peak amplitude and linewidth (defined asp1/ ffiffiffi cT2 and converted to peak-to-peak values by the classical 2= 3 factor) were deduced from the fit.
Fig. 1. Saturation of the ESR signal of an irradiated LiF sample containing high purity Li colloids. Discrete symbols describe experiments: line amplitude (dots) and peak-to-peak linewidth (triangles). Power units are given in mW (bottom scale) and in W (top scale). Lines come from a three-lorentzian fit of the data.
47
In Fig. 1 we give the variation of the amplitude and the peak-topeak linewidth versus the square root of the power. The microwave power was taken from the attenuator setting, assuming that 0 dB corresponds to 200 mW. In such a presentation, the amplitude is proportional to the abscissa in the absence of saturation, which is the case here for P < 1 mW. In order to analyze these results, we want to compare them with the classical saturation theory, described for instance by Poole [10], see his Fig. 18-5. The saturation factor s is given by:
s¼
1 1 þ 14 B21 c2 T 1 T 2
:
ð1Þ
The linewidth is given by:
DB1=2 ¼ DB01=2 s1=2 with DB01=2 ¼ 1=cT 2 ;
ð2Þ
and the peak-to-peak amplitude by: 3=2 y0m ¼ B1 y00 : m s
ð3Þ
We notice that some authors do not write the 1/4 factor appearing in the denominator of Eq. (1): Poole uses for B1 the linear field amplitude present in a rectangular TE cavity, whereas other authors use the rotating field amplitude, which is half the preceding one. Poole and Farach [11] (p. 8) discuss this point in detail. The formulas given by Poole give only a rough qualitative match to our experimental data and do not describe well the saturation at high power. However, for a first approach, let us consider the maximum amplitude in Fig. 1 and suppose that T1 = T2: we can then analyze the saturation data. It is well known that T1 = T2 for metals, according to Pines and Slichter [12], because in a metal electrons are scattered very fast all around the Fermi surface: the correlation time, of the order of 1014 s, is short compared with the Larmor period. In such a rough approach, P0, power for the maximum amplipffiffiffi tude, is 2.25 mW. For this power, B1 ¼ 1=ðcT 2 2Þ ¼ 6 lT and we get B1 = 57 lT for the nominal P (200 mW). Lund et al. [13] point out the importance of checking that we are in the slow passage regime: their Eq. (9) demands that xmBm kL/T1, where xm and Bm are the modulation angular frequency and amplitude and kL the lorentzian linewidth. Here we take kL = 10 lT, Bm = 2 lT, xm = 6 105 s1 and T1 = T2 = 1/cDB = 6.5 107 s. It gives 1.2 < 15 T s1, verifying the slow passage condition. On passage effects, one can refer to the book by Eaton et al. [14], pp. 54–55, who give another slow-passage condition, fm1 T 1 , which is also fulfilled here. In order to get a more quantitative analysis, it is tempting to use the model proposed by Lund et al. [13]. They describe the saturation behaviour of an inhomogeneously broadened ESR line modeled by a gaussian envelope of lorentzian spin packets. We use here their model, which is quite simple to compute. In their notations, for given values of the power P, the linewidths DBL and DBG, we calculate a, t, r and z. It is then necessary to compute the complex error function w(z), for which we use the computer procedure described by Abrarov and Quine [15] (Eq. (14)). pffiffiffi We take the real part of w, u, we get the signal which writes u P =t, and finally calculate the amplitude and width of the derivative of this signal. By using this model, we get a satisfactory agreement with the slowly decreasing amplitude with power only by putting in the model a high gaussian/lorentzian ratio, typically DBG 5 DBL. In these conditions, the linewidth of the signal varies only little with power, in contradiction with experiment. We remark also that the experimental signal line shape is quasi-lorentzian, and does not show the Voigt-type shape predicted by Lund’s model. We are then forced to abandon this model. We know from our experiments that there is a size distribution of the Li colloids in our samples. Such a distribution was unambiguously observed in former experiments on irradiated lithium oxide
48
F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
(Li2O) [16] and was highly suspected in the case of LiF [1]. The model proposed here is based on the hypothesis that there is some distribution of colloid sizes in the sample, which causes a distribution for linewidth and saturation power. In order to keep things simple, we try to describe this distribution with a sum of three lorentzian lines; this choice is quite arbitrary, but with only two lines we could not get a satisfactory fit. We do not believe that these three lines have a precise physical meaning, but that they are a simple model scheme for describing the colloid size distribution. These three lorentzian lines are taken with the same g-factor, 2.0023. We get a priori nine adjustable parameters: three saturation powers P01, P02, P03, three linewidths D1, D2, D3 and three intensity coefficients. Remembering that T1 = T2, P values and D are linked, because one can write P 01 =D21 ¼ P 02 =D22 ¼ P03 =D23 , which leaves us with seven parameters. We adjust these parameters for the amplitude and the linewidth versus microwave power, and get a rather satisfactory fit, see Fig. 1. The parameters used here include linewidth values of 6.9, 19.1 and 45.9 lT. From this fit we can deduce B1 = 51.5 lT in our cavity at full power (200 mW), a value not very different from the value given above from an oversimplified model. This value can be compared also to the factor 4.4 lT/ mW1/2 given by Lund [13] (in this paper, there is a misprint between gauss and mT units [17]) for a standard TE102 cavity with Q = 2500, which is 62.2 lT at 200 mW or 44 lT for a double volume TE104 cavity.
spectrum), there is an important component when the phase detection is in quadrature. We interpret these oscillations by the presence of interferences occurring when 1/T2 and fm are of the same order of magnitude, fm being the modulation frequency. These oscillations are of comparable nature with the so called ‘‘wiggles’’ observed in NMR and described by Abragam [19], pp. 85–86. They can be interpreted in the framework of the fast passage conditions ([14], p. 55). Stoner et al. [20] and Joshi et al. [21] describe experiments of direct-detected rapid-scan EPR showing well resolved wiggles; their interpretation in terms of Bloch equations will be referred to in the following. In our case, we see interferences between successive stimulations of the spin resonance. For the sake of simplicity, let us consider that the ESR signal is of negligible width relatively to the modulation width, which is the case in extreme overmodulation conditions. Then, the resonance is excited twice in every modulation period, and interferences can occur between these successive resonance excitations. In this framework, the simplest model comes from calculating the interferences between the two signal excitations closest in time in the same modulation period. A straightforward calculation gives:
4. Oscillations under extreme overmodulation
where k is an integer indexing an oscillation, Bm is the modulation amplitude, xm is the modulation angular frequency (xm = 2pfm) and R = B/Bm is the field value (taken from the center of the resonance) divided by the modulation. For each k integer value, the position of the kth peak is easily computed by numerically solving Eq. (4) for R, i.e. the field B. This very simple model accounts well for the peak positions and their dependence on modulation amplitude and frequency, when looking in the outer part of the spectra. There are however some discrepancies near the middle of the spectra, where oscillations are not very regular: such irregularities are seen more easily on the spectra in quadrature, such as the third spectrum in Fig. 2a. This is quite easily understood if we consider that near B = 0, which corresponds to the middle of the spectrum, we can no more suppose that only two excitations of the resonance are involved. Taking into account a third excitation, from the next modulation period, we can describe satisfactorily the irregularities of the oscillations near the spectrum center: see Fig. 3, where experiment
It is well known that, under extreme overmodulation conditions, the shape of an ESR line changes drastically, with a peakto-peak linewidth equal to the modulation amplitude: see for instance Weil et al. [18] p 504. We have observed the signal from one of our irradiated LiF samples under such conditions. In Fig. 2a we show overmodulated spectra, taken at 100 kHz, except the second spectrum taken at 50 kHz. The modulation amplitude was 0.4 mT peak-to-peak, except the fourth spectrum taken at 5 lT peak-to-peak. For the two upper spectra the phase of the phase sensitive detector was 0°, and it was 90° for the two others. On the first three spectra, oscillations are seen, and are even more visible in the spectrum taken in quadrature detection. Oscillations are closer at lower modulation frequency; their behaviour do not depend on the magnetic field sweep speed. Even when the signal is not overmodulated (last
(a)
k¼
cBm pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 R 1 R ðp 2 arcsin RÞ 2 p xm
ð4Þ
(b)
Fig. 2. (a) experimental ESR signals taken under extreme overmodulation conditions for the three upper spectra: 0.4 mT peak-to-peak modulation – modulation frequency 100 or 50 kHz – detection phase 0 or 90°. Last spectrum taken at 90° without overmodulation. (b) spectra calculated from Bloch equations in the same conditions. We took T1 = T2 = 2 ls.
F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
Fig. 3. Details for a spectrum of Fig. 2, with 0.4 mT peak-to-peak modulation, 100 kHz modulation frequency and 90° detection phase. We show the experimental signal, the calculation from Bloch equations and from the simple interference model described in the text.
is compared with the present model – termed ‘‘simple phase-shift model’’ – and with a more quantitative calculation that is presented in the next paragraph. An independent model can be used. A similar approach was used in the past, for instance by Stoner et al. [20]. We start from Bloch equations written in the rotating frame, as given by Abragam [19], with his Eq. (14) p. 45:
dMx Mx þ DxM y ¼ dt T2 dMy My x1 M z ¼ DxM x dt T2 dMz Mz M0 ¼ x1 M y dt T1
ð5Þ
with Dx = x x0, x0 = cB0 and x1 = cB1. The static magnetic field is oriented along the z axis; Mx, My and Mz are the three components of the magnetization, and M0 the magnetization at equilibrium. In practice we take T1 = T2, as stated above. We introduce in B0 the field modulation and we solve the equation system (5) using the Runge–Kutta method, as described in Numerical Recipes [22]. After letting the system go to equilibrium, we integrate on one modulation period the My value multiplied by a factor proportional to the (possibly phase-shifted) modulation in order to simulate the phase
(a)
49
sensitive detection. In Figs. 2b and 3, we show the results of the simulations of the experimental spectra of Fig. 2a. It is quite clear that the model works very well and describes all the features of the oscillations, even near the center of the spectra. We add here a final note: another oscillatory effect, linked to sidebands, is referenced in literature but must not be confused with the present effect. This effect is described for instance in Weil et al. [18], p. 505; oscillations are also described by Kälin et al. [23], in the 500–1000 kHz modulation frequency range. In both cases the effects are reproduced with our simple Runge–Kutta program which solves Bloch equations. In the case of Kälin et al., we get a perfect reproduction of the totality of their figures, and of their Figs. 6 and 7 in particular: oscillations are periodic versus magnetic field, appearing at higher modulation frequency or longer T2, while the oscillations described in the present paper are not periodic. In fact, when performing a simulation for 300 kHz, the two types of oscillations are seen simultaneously.
5. Bistability The last phenomenon described in this paper is the appearance of bistability, which manifests itself in spectra becoming strongly different whether the magnetic field is swept upwards or downwards. In the past, Vigreux et al. [24] described a very interesting phenomenon of this kind, observed with lithium particles in Li2O powders irradiated by 1-MeV electrons. They termed it ‘‘bistable conduction electron spin resonance’’ and showed that it is due to a dynamic nuclear polarization occurring by the Overhauser effect when the ESR transition of the conduction electrons is saturated. In her thesis work, Vigreux [25] mentioned another bistability effect of a different nature, which we reproduced here. She showed that it was of instrumental origin but did not give a definitive explanation for it. In Fig. 4a, we show the results of experiments on an irradiated LiF sample with a size larger than the preceding ones. The two lines which show very steep features in their middle are taken with magnetic field increasing and decreasing. The third spectrum, with a more conventional shape, was taken in the same conditions except that the automatic frequency control (AFC) was switched off. Like Vigreux, we measured the microwave frequency during the field sweep and saw that the frequency value is affected by the same steep jump that the ESR signal. Moreover, Vigreux [25] states that after correcting the magnetic field by the frequency
(b)
Fig. 4. (a) experimental ESR signals showing a bistability behaviour: the two continuous lines show spectra taken with increasing or decreasing magnetic field; the dashed line is for a spectrum taken with AFC off. (b) simulation of the bistability for increasing magnetic field, with (continuous line) or without (dashed line) AFC.
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F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
shift, the lineshape returns to its normal shape, which proves that the phenomenon is purely instrumental. A similar phenomenon, with somewhat different effect on the EPR spectra, was previously reported by Ilangovan et al. [26] and called extremely narrow EPR signals. The artifact is discussed in Ref. [14], p. 74. We give here an explanation for the phenomenon. We believe that the bistability appears when the ESR resonance linewidth becomes narrower than the electrical resonance of the spectrometer cavity, if the two widths are given in the same suitable unit. Consider an ESR cavity working at X-band (f 9 GHz) with a quality factor Q = 3000. The width of its resonance is Df = f/Q 3 MHz. In terms of magnetic field, through the gyromagnetic ratio c, it corresponds to a width of about 0.1 mT. Then, in presence of an ESR signal narrower than this value, a narrow bump adds on the cavity mode. If this bump is intense enough, it gives rise to two minima on this mode. The AFC system insures that the spectrometer microwave source remains locked on the mode minimum, and in the presence of two minima, this mechanism is susceptible to induce bistability, because the chosen local minimum will depend on the field sweep history. As an illustration of this idea, we simulated the bistable behaviour by a very simple mode shape, taken arbitrarily as lorentzian, whose depth D is written as:
D¼
C S þ f 2 þ Df 2 ðf f0 Þ2 þ DB2
ð6Þ
where C S describe respectively the depth of the cavity mode and the ESR signal absorption, f is the cavity resonance frequency without ESR and f0 describes, in frequency units, the sweeping of the magnetic field; Df is the cavity resonance width and DB the signal width, always in frequency units. We calculate the field and amplitude of the local minimum of this function when the field is swept and take the derivative of the amplitude in order to simulate experiments. For a certain value of the field, there is a brutal change in the signal because the local minimum disappears, causing the minimum value to shift at once. This is shown in Fig. 4b, where one line corresponds to the simulated lorentzian signal in the absence of AFC, and the other shows the discontinuity appearing with increasing field when the AFC mechanism is operating. 6. Conclusion We have shown in this paper some interesting phenomena which can be observed on ESR lines when the linewidth becomes uncommonly narrow. Some of them can give rise to calibration procedures, such as the saturation (used to calibrate the rf magnetic field) or the overmodulation (to calibrate the field modulation amplitude), while others are spectacular and purely instrumental effects.
References [1] F. Beuneu, P. Vajda, O.J. Zogal, Li colloids created by electron-irradiation of LiF: a great wealth of properties, Nucl. Instrum. Methods Phys. Res. B 191 (2002) 149–153. [2] M. Lambert, Ch. Mazières, A. Guinier, Précipitation de lithium dans les monocristaux de fluorure de lithium irradiés aux neutrons thermiques, J. Phys. Chem. Solids 18 (1961) 129–138. [3] R. Kaplan, P.J. Bray, Electron-spin paramagnetic resonance studies of neutronirradiated LiF, Phys. Rev. 129 (1963) 1919–1935. [4] A. Stesmans, Y. Wu, Evidence for a phase-transition-induced change in the surface spin-flip probability of conduction electrons from CESR on n-irradiated LiF; its application as an intensity reference, J. Phys. D: Appl. Phys. 21 (1988) 1205–1214. [5] S.G. L’vov, F.G. Cherkasov, A.Ya. Vitol, V.A. Silaev, ESR and ESR-imaging of heavily irradiated alkali halide crystals, Appl. Radiat. Isot. 47 (1996) 1615– 1619. [6] A. Van den Bosch, c-Radiolysis of LiF, Radiat. Eff. 19 (1973) 129–133. [7] F. Beuneu, P. Monod, The Elliott relation in pure metals, Phys. Rev. B 18 (1978) 2422–2425. [8] P. Monod, F. Beuneu, Conduction-electron spin flip by phonons in metals: analysis of experimental data, Phys. Rev. B 19 (1979) 911–916. [9] N.S. VanderVen, Measurement of the g value of conduction electrons in lithium metal, Phys. Rev. 168 (1968) 787–795. [10] C.P. Poole, Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques, Interscience Publishers, 1967. [11] C.P. Poole, H.A. Farach, Relaxation in Magnetic Resonance: Dielectric and Mössbauer Applications, Academic Press, 1971. [12] D. Pines, C.P. Slichter, Relaxation times in magnetic resonance, Phys. Rev. 100 (1955) 1014–1020. [13] A. Lund, E. Sagstuen, A. Sanderud, J. Maruani, Relaxation-time determination from continuous-microwave saturation of EPR spectra, Radiat. Res. 172 (2009) 753–760. [14] G.R. Eaton, S.S. Eaton, D.P. Barr, R.T. Weber, Quantitative EPR, Springer, Wien, New York, 2010. [15] S.M. Abrarov, B.M. Quine, Efficient algorithmic implementation of the Voigt/ complex error function based on exponential series approximation, Appl. Math. Comput. 218 (2011) 1894–1902. [16] F. Beuneu, P. Vajda, Spectroscopic evidence for large (1 lm) lithium-colloid creation in electron-irradiated Li2O single crystals, Phys. Rev. Lett. 76 (1996) 4544–4547. [17] A. Lund, E. Sagstuen, Private communication, 2012. [18] J.A. Weil, J.R. Bolton, J.E. Wertz, Electron Paramagnetic Resonance: Elementary Theory and Practical Applications, John Wiley & Sons, 1994. [19] A. Abragam, The Principles of Nuclear Magnetism, Oxford at the Clarendon Press, 1961. [20] J.W. Stoner, D. Szymanski, S.S. Eaton, R.W. Quine, G.A. Rinard, G.R. Eaton, Direct-detected rapid-scan EPR at 250 MHz, J. Magn. Reson. 170 (2004) 127– 135. [21] J.P. Joshi, G.R. Eaton, S.S. Eaton, Impact of resonator on direct-detected rapidscan EPR at 9.8 GHz, Appl. Magn. Reson. 28 (2005) 239–249. [22] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, 1988. [23] M. Kälin, I. Gromov, A. Schweiger, The continuous wave electron paramagnetic resonance experiment revisited, J. Magn. Reson. 160 (2003) 166–182. [24] C. Vigreux, L. Binet, D. Gourier, Bistable conduction electron spin resonance in metallic lithium particles, J. Phys. Chem. B 102 (1998) 1176–1181. [25] C. Vigreux, Thesis in Physics, Paris VI University, 1999. [26] G. Ilangovan, J.L. Zweier, P. Kuppusamy, Electrochemical preparation and EPR studies of lithium phthalocyanine. Part 2: Particle-size-dependent line broadening by molecular oxygen and its implications as an oximetry probe, J. Phys. Chem. B 104 (2000) 9404–9410.
Contents lists available at SciVerse ScienceDirect
Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Unusual behaviour of a very narrow electron spin resonance signal François Beuneu Laboratoire des Solides Irradiés, CNRS-CEA, École Polytechnique, F-91128 Palaiseau, France
a r t i c l e
i n f o
Article history: Received 10 October 2012 Revised 7 November 2012 Available online 23 November 2012 Keywords: A. Lithium metal colloids B. Electron spin resonance
a b s t r a c t Unusually narrow electron spin resonance lines can give birth to spectacular and interesting phenomena. We use here very pure lithium metal colloids created in electron-irradiated LiF, giving rise to a very narrow Li metal line. In these samples, three interesting phenomena are discussed and interpreted: signal saturation in particularly good conditions; oscillations appearing on strongly overmodulated lines; signal bistability, when the signal shape seems to depend on the sample magnetic history. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The present paper describes some interesting phenomena observed in electron spin resonance (ESR) experiments on samples giving very narrow resonance lines, i.e. linewidths lower than 0.1G = 10 lT. Among materials giving birth to such narrow lines, we chose to work with Li metal small clusters or colloids embedded in an insulating matrix, which were obtained by electron-irradiation of single crystals of lithium fluoride LiF. In the past, we made a detailed study of these colloids with ESR [1]. In the past, the creation of lithium precipitates in LiF by irradiation was initiated by Lambert et al. [2], who used thermal neutron irradiations. Such irradiations were also done by Kaplan and Bray [3] and by Stesmans and Wu [4]. Electron irradiation was used by L’vov et al. [5]. Even with pure electronic excitations, as with c-ray irradiation, Van den Bosch [6] reported the observation of Li metal in LiF, but unfortunately did not do ESR experiments. Under room-temperature irradiation, fluorine vacancies are created and group themselves into anion-vacancies clusters, which therefore contain only Li atoms. Depending on the electron flux and fluence, these clusters can grow and reach sizes of several hundreds of nm, and even in the lm range. These very pure lithium metal precipitates are ideal candidates for giving extremely narrow ESR lines coming from conduction electrons of the metal. Conduction electron spin resonance (CESR) was extensively studied in the past and is generally a phenomenon difficult to detect, and in most cases only near liquid helium temperature. We gave a review of CESR properties in metals in two papers [7,8]: Li is quite peculiar because spin–orbit interaction is very small in this light metal. This leads to a narrow CESR line, without temperature variation of the linewidth, a unique behaviour. We recall that the signal intensity is independent of temperature, like for E-mail address: [email protected] 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.11.011
every metal, because it is proportional to the paramagnetic susceptibility which is of Pauli-type for conduction electrons. The spin–orbit weakness is also responsible for the fact that the g-factor of Li conduction electrons is extremely close to the freeelectron value, g 2.0023. Indeed, VanderVen [9] made precise measurements of the Li g-factor and stated that ‘‘the shift of the conduction–electron g value relative to that of the free electron is dg = (6.1 ± 0.2) 106.’’ A simple application of Li-colloids samples in ESR is to use them as g-factor standards, because their g is precisely known and their narrow width enables a precise determination of the resonance position. After describing the experimental conditions (Section 2) and particularly the way samples are produced, the paper describes three interesting phenomena in which the narrow ESR linewidth is crucial: a precise study of line saturation (Section 3) with application to rf magnetic field measurement in the ESR cavity; under extreme overmodulation conditions, oscillations are observed on the spectra (Section 4) and an interpretation is given; in the case of large samples, spectra become different if magnetic field is increased or decreased (Section 5): this bistability is studied and interpreted as a purely instrumental effect. 2. Experimental Specimens of LiF of dimensions 3–4 mm2, 500–1000 lm thick, were prepared by cleaving LiF crystals along (1 0 0). They were then uniformly irradiated by 2.5-MeV electrons from our Van de Graaff accelerator. During electron bombardment, samples were fixed on a water-cooled copper target with a 10-lm-thick copper foil. The irradiation temperature, monitored by a thermocouple, was about 50 °C. The introduced Li colloids were then detected and characterised by CESR, with a Bruker ER 200D or EMX X-band spectrometer, in a rectangular double cavity (TE104 mode) with a quality factor near 3000.
F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
In Ref. [1], we described two strategies to get extremely narrow Li ESR lines. The first one consists in using strong electron flux and fluence (for instance 3.6 1019 e/cm2 = 5.8 C/cm2 at 40 lA/cm2): the obtained lorentzian line is directly very intense and narrow (610 lT). The second uses lower irradiation conditions, such as 1.6 1019 e/cm2 = 2.6 C/cm2 at 6 lA/cm2. The line is then quite broad, but after an annealing near 700 °C, typical linewidth values are very small, about 3–4 lT (30–40 mG). Three samples are used in the present study. The first one, used in Section 3, was submitted to a fluence of 2.33 C/cm2 and a beam current of 16 lA/cm2. The second one, used in Section 4, received 1.17 C/cm2 under 15 lA/cm2. These two samples were of the second type described above: they gave broad CESR lines (typically 0.1 mT) after irradiation and were annealed at high temperature (700 °C); after annealing, they fell into small pieces, with a typical volume about 1 mm3, and one piece for each sample was used for the experiments described here. The third sample, used in Section 5, was irradiated at 28 lA/cm2 and 1.81 C/cm2. It was bigger in size, with a weight of 2.5 mg, and gave a narrow line (9 lT) without annealing. Samples of the same batch than the two first samples were measured at variable temperature, from 4 K to 300 K. Their width and intensity did not vary with temperature. For all samples used here, the lineshape was symmetrical and generally lorentzian. The dysonian lineshape very often observed in CESR is due to an incomplete microwave penetration into the sample because of skin effect. In our case, at room temperature, the skin depth is larger than the Li particle size, so that no dysonian shape is observed.
3. Saturation A saturation ESR experiment was performed at room temperature on the first irradiated LiF sample. The maximum source power was 200 mW, and the microwave attenuator was varied from 0 to 50 dB. The modulation was set at 100 kHz frequency and 2 lT peak-to-peak amplitude. The ESR line was adjusted numerically with a lorentzian shape by a least-squares program and the peak-to-peak amplitude and linewidth (defined asp1/ ffiffiffi cT2 and converted to peak-to-peak values by the classical 2= 3 factor) were deduced from the fit.
Fig. 1. Saturation of the ESR signal of an irradiated LiF sample containing high purity Li colloids. Discrete symbols describe experiments: line amplitude (dots) and peak-to-peak linewidth (triangles). Power units are given in mW (bottom scale) and in W (top scale). Lines come from a three-lorentzian fit of the data.
47
In Fig. 1 we give the variation of the amplitude and the peak-topeak linewidth versus the square root of the power. The microwave power was taken from the attenuator setting, assuming that 0 dB corresponds to 200 mW. In such a presentation, the amplitude is proportional to the abscissa in the absence of saturation, which is the case here for P < 1 mW. In order to analyze these results, we want to compare them with the classical saturation theory, described for instance by Poole [10], see his Fig. 18-5. The saturation factor s is given by:
s¼
1 1 þ 14 B21 c2 T 1 T 2
:
ð1Þ
The linewidth is given by:
DB1=2 ¼ DB01=2 s1=2 with DB01=2 ¼ 1=cT 2 ;
ð2Þ
and the peak-to-peak amplitude by: 3=2 y0m ¼ B1 y00 : m s
ð3Þ
We notice that some authors do not write the 1/4 factor appearing in the denominator of Eq. (1): Poole uses for B1 the linear field amplitude present in a rectangular TE cavity, whereas other authors use the rotating field amplitude, which is half the preceding one. Poole and Farach [11] (p. 8) discuss this point in detail. The formulas given by Poole give only a rough qualitative match to our experimental data and do not describe well the saturation at high power. However, for a first approach, let us consider the maximum amplitude in Fig. 1 and suppose that T1 = T2: we can then analyze the saturation data. It is well known that T1 = T2 for metals, according to Pines and Slichter [12], because in a metal electrons are scattered very fast all around the Fermi surface: the correlation time, of the order of 1014 s, is short compared with the Larmor period. In such a rough approach, P0, power for the maximum amplipffiffiffi tude, is 2.25 mW. For this power, B1 ¼ 1=ðcT 2 2Þ ¼ 6 lT and we get B1 = 57 lT for the nominal P (200 mW). Lund et al. [13] point out the importance of checking that we are in the slow passage regime: their Eq. (9) demands that xmBm kL/T1, where xm and Bm are the modulation angular frequency and amplitude and kL the lorentzian linewidth. Here we take kL = 10 lT, Bm = 2 lT, xm = 6 105 s1 and T1 = T2 = 1/cDB = 6.5 107 s. It gives 1.2 < 15 T s1, verifying the slow passage condition. On passage effects, one can refer to the book by Eaton et al. [14], pp. 54–55, who give another slow-passage condition, fm1 T 1 , which is also fulfilled here. In order to get a more quantitative analysis, it is tempting to use the model proposed by Lund et al. [13]. They describe the saturation behaviour of an inhomogeneously broadened ESR line modeled by a gaussian envelope of lorentzian spin packets. We use here their model, which is quite simple to compute. In their notations, for given values of the power P, the linewidths DBL and DBG, we calculate a, t, r and z. It is then necessary to compute the complex error function w(z), for which we use the computer procedure described by Abrarov and Quine [15] (Eq. (14)). pffiffiffi We take the real part of w, u, we get the signal which writes u P =t, and finally calculate the amplitude and width of the derivative of this signal. By using this model, we get a satisfactory agreement with the slowly decreasing amplitude with power only by putting in the model a high gaussian/lorentzian ratio, typically DBG 5 DBL. In these conditions, the linewidth of the signal varies only little with power, in contradiction with experiment. We remark also that the experimental signal line shape is quasi-lorentzian, and does not show the Voigt-type shape predicted by Lund’s model. We are then forced to abandon this model. We know from our experiments that there is a size distribution of the Li colloids in our samples. Such a distribution was unambiguously observed in former experiments on irradiated lithium oxide
48
F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
(Li2O) [16] and was highly suspected in the case of LiF [1]. The model proposed here is based on the hypothesis that there is some distribution of colloid sizes in the sample, which causes a distribution for linewidth and saturation power. In order to keep things simple, we try to describe this distribution with a sum of three lorentzian lines; this choice is quite arbitrary, but with only two lines we could not get a satisfactory fit. We do not believe that these three lines have a precise physical meaning, but that they are a simple model scheme for describing the colloid size distribution. These three lorentzian lines are taken with the same g-factor, 2.0023. We get a priori nine adjustable parameters: three saturation powers P01, P02, P03, three linewidths D1, D2, D3 and three intensity coefficients. Remembering that T1 = T2, P values and D are linked, because one can write P 01 =D21 ¼ P 02 =D22 ¼ P03 =D23 , which leaves us with seven parameters. We adjust these parameters for the amplitude and the linewidth versus microwave power, and get a rather satisfactory fit, see Fig. 1. The parameters used here include linewidth values of 6.9, 19.1 and 45.9 lT. From this fit we can deduce B1 = 51.5 lT in our cavity at full power (200 mW), a value not very different from the value given above from an oversimplified model. This value can be compared also to the factor 4.4 lT/ mW1/2 given by Lund [13] (in this paper, there is a misprint between gauss and mT units [17]) for a standard TE102 cavity with Q = 2500, which is 62.2 lT at 200 mW or 44 lT for a double volume TE104 cavity.
spectrum), there is an important component when the phase detection is in quadrature. We interpret these oscillations by the presence of interferences occurring when 1/T2 and fm are of the same order of magnitude, fm being the modulation frequency. These oscillations are of comparable nature with the so called ‘‘wiggles’’ observed in NMR and described by Abragam [19], pp. 85–86. They can be interpreted in the framework of the fast passage conditions ([14], p. 55). Stoner et al. [20] and Joshi et al. [21] describe experiments of direct-detected rapid-scan EPR showing well resolved wiggles; their interpretation in terms of Bloch equations will be referred to in the following. In our case, we see interferences between successive stimulations of the spin resonance. For the sake of simplicity, let us consider that the ESR signal is of negligible width relatively to the modulation width, which is the case in extreme overmodulation conditions. Then, the resonance is excited twice in every modulation period, and interferences can occur between these successive resonance excitations. In this framework, the simplest model comes from calculating the interferences between the two signal excitations closest in time in the same modulation period. A straightforward calculation gives:
4. Oscillations under extreme overmodulation
where k is an integer indexing an oscillation, Bm is the modulation amplitude, xm is the modulation angular frequency (xm = 2pfm) and R = B/Bm is the field value (taken from the center of the resonance) divided by the modulation. For each k integer value, the position of the kth peak is easily computed by numerically solving Eq. (4) for R, i.e. the field B. This very simple model accounts well for the peak positions and their dependence on modulation amplitude and frequency, when looking in the outer part of the spectra. There are however some discrepancies near the middle of the spectra, where oscillations are not very regular: such irregularities are seen more easily on the spectra in quadrature, such as the third spectrum in Fig. 2a. This is quite easily understood if we consider that near B = 0, which corresponds to the middle of the spectrum, we can no more suppose that only two excitations of the resonance are involved. Taking into account a third excitation, from the next modulation period, we can describe satisfactorily the irregularities of the oscillations near the spectrum center: see Fig. 3, where experiment
It is well known that, under extreme overmodulation conditions, the shape of an ESR line changes drastically, with a peakto-peak linewidth equal to the modulation amplitude: see for instance Weil et al. [18] p 504. We have observed the signal from one of our irradiated LiF samples under such conditions. In Fig. 2a we show overmodulated spectra, taken at 100 kHz, except the second spectrum taken at 50 kHz. The modulation amplitude was 0.4 mT peak-to-peak, except the fourth spectrum taken at 5 lT peak-to-peak. For the two upper spectra the phase of the phase sensitive detector was 0°, and it was 90° for the two others. On the first three spectra, oscillations are seen, and are even more visible in the spectrum taken in quadrature detection. Oscillations are closer at lower modulation frequency; their behaviour do not depend on the magnetic field sweep speed. Even when the signal is not overmodulated (last
(a)
k¼
cBm pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 R 1 R ðp 2 arcsin RÞ 2 p xm
ð4Þ
(b)
Fig. 2. (a) experimental ESR signals taken under extreme overmodulation conditions for the three upper spectra: 0.4 mT peak-to-peak modulation – modulation frequency 100 or 50 kHz – detection phase 0 or 90°. Last spectrum taken at 90° without overmodulation. (b) spectra calculated from Bloch equations in the same conditions. We took T1 = T2 = 2 ls.
F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
Fig. 3. Details for a spectrum of Fig. 2, with 0.4 mT peak-to-peak modulation, 100 kHz modulation frequency and 90° detection phase. We show the experimental signal, the calculation from Bloch equations and from the simple interference model described in the text.
is compared with the present model – termed ‘‘simple phase-shift model’’ – and with a more quantitative calculation that is presented in the next paragraph. An independent model can be used. A similar approach was used in the past, for instance by Stoner et al. [20]. We start from Bloch equations written in the rotating frame, as given by Abragam [19], with his Eq. (14) p. 45:
dMx Mx þ DxM y ¼ dt T2 dMy My x1 M z ¼ DxM x dt T2 dMz Mz M0 ¼ x1 M y dt T1
ð5Þ
with Dx = x x0, x0 = cB0 and x1 = cB1. The static magnetic field is oriented along the z axis; Mx, My and Mz are the three components of the magnetization, and M0 the magnetization at equilibrium. In practice we take T1 = T2, as stated above. We introduce in B0 the field modulation and we solve the equation system (5) using the Runge–Kutta method, as described in Numerical Recipes [22]. After letting the system go to equilibrium, we integrate on one modulation period the My value multiplied by a factor proportional to the (possibly phase-shifted) modulation in order to simulate the phase
(a)
49
sensitive detection. In Figs. 2b and 3, we show the results of the simulations of the experimental spectra of Fig. 2a. It is quite clear that the model works very well and describes all the features of the oscillations, even near the center of the spectra. We add here a final note: another oscillatory effect, linked to sidebands, is referenced in literature but must not be confused with the present effect. This effect is described for instance in Weil et al. [18], p. 505; oscillations are also described by Kälin et al. [23], in the 500–1000 kHz modulation frequency range. In both cases the effects are reproduced with our simple Runge–Kutta program which solves Bloch equations. In the case of Kälin et al., we get a perfect reproduction of the totality of their figures, and of their Figs. 6 and 7 in particular: oscillations are periodic versus magnetic field, appearing at higher modulation frequency or longer T2, while the oscillations described in the present paper are not periodic. In fact, when performing a simulation for 300 kHz, the two types of oscillations are seen simultaneously.
5. Bistability The last phenomenon described in this paper is the appearance of bistability, which manifests itself in spectra becoming strongly different whether the magnetic field is swept upwards or downwards. In the past, Vigreux et al. [24] described a very interesting phenomenon of this kind, observed with lithium particles in Li2O powders irradiated by 1-MeV electrons. They termed it ‘‘bistable conduction electron spin resonance’’ and showed that it is due to a dynamic nuclear polarization occurring by the Overhauser effect when the ESR transition of the conduction electrons is saturated. In her thesis work, Vigreux [25] mentioned another bistability effect of a different nature, which we reproduced here. She showed that it was of instrumental origin but did not give a definitive explanation for it. In Fig. 4a, we show the results of experiments on an irradiated LiF sample with a size larger than the preceding ones. The two lines which show very steep features in their middle are taken with magnetic field increasing and decreasing. The third spectrum, with a more conventional shape, was taken in the same conditions except that the automatic frequency control (AFC) was switched off. Like Vigreux, we measured the microwave frequency during the field sweep and saw that the frequency value is affected by the same steep jump that the ESR signal. Moreover, Vigreux [25] states that after correcting the magnetic field by the frequency
(b)
Fig. 4. (a) experimental ESR signals showing a bistability behaviour: the two continuous lines show spectra taken with increasing or decreasing magnetic field; the dashed line is for a spectrum taken with AFC off. (b) simulation of the bistability for increasing magnetic field, with (continuous line) or without (dashed line) AFC.
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F. Beuneu / Journal of Magnetic Resonance 227 (2013) 46–50
shift, the lineshape returns to its normal shape, which proves that the phenomenon is purely instrumental. A similar phenomenon, with somewhat different effect on the EPR spectra, was previously reported by Ilangovan et al. [26] and called extremely narrow EPR signals. The artifact is discussed in Ref. [14], p. 74. We give here an explanation for the phenomenon. We believe that the bistability appears when the ESR resonance linewidth becomes narrower than the electrical resonance of the spectrometer cavity, if the two widths are given in the same suitable unit. Consider an ESR cavity working at X-band (f 9 GHz) with a quality factor Q = 3000. The width of its resonance is Df = f/Q 3 MHz. In terms of magnetic field, through the gyromagnetic ratio c, it corresponds to a width of about 0.1 mT. Then, in presence of an ESR signal narrower than this value, a narrow bump adds on the cavity mode. If this bump is intense enough, it gives rise to two minima on this mode. The AFC system insures that the spectrometer microwave source remains locked on the mode minimum, and in the presence of two minima, this mechanism is susceptible to induce bistability, because the chosen local minimum will depend on the field sweep history. As an illustration of this idea, we simulated the bistable behaviour by a very simple mode shape, taken arbitrarily as lorentzian, whose depth D is written as:
D¼
C S þ f 2 þ Df 2 ðf f0 Þ2 þ DB2
ð6Þ
where C S describe respectively the depth of the cavity mode and the ESR signal absorption, f is the cavity resonance frequency without ESR and f0 describes, in frequency units, the sweeping of the magnetic field; Df is the cavity resonance width and DB the signal width, always in frequency units. We calculate the field and amplitude of the local minimum of this function when the field is swept and take the derivative of the amplitude in order to simulate experiments. For a certain value of the field, there is a brutal change in the signal because the local minimum disappears, causing the minimum value to shift at once. This is shown in Fig. 4b, where one line corresponds to the simulated lorentzian signal in the absence of AFC, and the other shows the discontinuity appearing with increasing field when the AFC mechanism is operating. 6. Conclusion We have shown in this paper some interesting phenomena which can be observed on ESR lines when the linewidth becomes uncommonly narrow. Some of them can give rise to calibration procedures, such as the saturation (used to calibrate the rf magnetic field) or the overmodulation (to calibrate the field modulation amplitude), while others are spectacular and purely instrumental effects.
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