Eur. Phys. J. D 51, 357–367 (2009) DOI: 10.1140/epjd/e2009-00027-7
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Spectral and temporal features of multiple spontaneous NMR-maser emissions D.J.-Y. Marion, P. Berthault, and H. Desvauxa CEA, IRAMIS, Service de Chimie Mol´eculaire, Laboratoire Structure et Dynamique par R´esonance Magn´etique, URA CEA/CNRS 331, 91191 Gif-sur-Yvette, France Received 10 July 2008 / Received in final form 25 September 2008 c EDP Sciences, Societ` Published online 6 February 2009 – a Italiana di Fisica, Springer-Verlag 2009 Abstract. When a system composed of dissolved laser-polarized xenon with negative spin temperature is put inside a high field NMR magnet, a series of spontaneous maser emissions can be observed. We report here their spectral and temporal features using a processing model derived from the solution of the Bloch equations in the presence of radiation damping. We show, in particular, that by combining Fourier transformation and squared modulus, a parameter allowing the characterization of the burst of transverse magnetization can be determined. This parameter is shown to be correlated with the radiated energy. Moreover, this processing clearly reveals features which can probably be assigned to effects resulting from distant dipolar fields. Finally, the analysis of the experimental data reveals an unexpected behavior of the 129 Xe transverse self-relaxation. PACS. 84.40.Ik Masers; gyrotrons – 82.56.-b Nuclear magnetic resonance – 76.60.Es Relaxation effects – 32.80.Xx Level crossing and optical pumping of atoms
1 Introduction The low sensitivity inherent to liquid-state NMR spectroscopy drives perpetually renewed interest for the development of polarized nuclear spin systems [1–3] used as a source of polarization for other nuclear spins [4–6]. Nevertheless, these solutions require the ability to manage highly polarized nuclear spin systems, which, due to (i) the intermolecular long-distance dipolar couplings (distant dipolar fields, DDF) [7–10], (ii) the breakdown of the high temperature approximation which requires the consideration of intermolecular coherences in the density matrix expression (low spin temperature) [11,12] and (iii) non-linear coupling between the magnetization and the coil (radiation damping, RD) [13–15] leads to the appearance of new physical phenomena [7,8,16–21], the description of which usually reveals the presence of chaos [22–25]. All these features originate from collective spin dynamics, a phenomenon usually neglected in liquids [26]. In this field, we have recently reported the unexpected observation of spontaneous multiple maser emissions [27]. Indeed, for samples containing dissolved laser-polarized xenon of magnetization anti-parallel to the magnetic field, a series of rf bursts – and not only one – is detected without any renewal of the xenon magnetization inside the detection region by transport mechanisms. It appears in the experiments that not all detectable fluctuations are able to grow up sufficiently to produce a maser emission, or that a
e-mail:
[email protected]
starting with “identical” conditions, the final spatial organization of the magnetization fully differs, all features revealing the chaotic nature of the phenomenon. Moreover, we have shown that the spin dynamics between two consecutive masers should ensure a reorganization of the xenon magnetization capable of triggering the subsequent emission [27]. The origin of these multiple emissions remains unclear since beside collective effects due to DDF, it might also be explained by chaos resulting from the BlochMaxwell equations [28] as in chaotic lasers at quasi-optical wavelengths [29]. The aim of the present paper is not to present a thorough study of the chaotic behavior of the system, but to rationalize our observations and to allow easier confrontations to results derived from numerical simulations based on DDF [22,24,30] or super-radiance approaches [31]. We are analyzing herein temporal and spectral features of different multiple maser emissions obtained in comparable experiments. Among the whole set of processing methods such as sliding-windows Fourier transformation, wavelets, phase analysis. . . , the absence of theoretical models which would have allowed the physical characterization of the deduced parameters, has driven us to the simplest one, which is based on Fourier transformation of the squared modulus of the signal. We show in particular that this processing scheme allows the extraction of a parameter characterizing the energy spent by each maser emission. This model also reveals the complexity of the spin dynamics in this type of polarized systems illustrated by unexpected values of the xenon transverse self-relaxation rates.
358
The European Physical Journal D
2 Theoretical framework 2.1 Generalized Bloch equations Considering a magnetically-oriented point of view, the dynamics of the spin system can be based on a classical description of the distribution of the xenon magnetization, M (r, t), over the sample [26]. M (r, t) is defined here as the sum of the nuclear magnetic moments present in a little volume dV around r. The dipolar interaction between a given region and the rest of the sample is taken into account in the generalized Bloch equations with help of the formalism of the average dipolar field Bd acting on M (r, t) and defined as: M (r , t) − 3(M (r , t) · u)u 3 μ0 d r (1) Bd (r, t) = 4π ||r − r||3 where u = (r − r)/||r − r||. This term being a perturbation to the Zeeman Hamiltonian in our conditions, only the projection of the average dipolar field on the static magnetic field direction has an influence on the spin dynamics. As stated before, the RD effect is a nonlinear feedback: the precessing magnetization creates a current in the coil; this current in turn creates a radio-frequency field which interacts with the magnetization. It must therefore be included in the modified Bloch equations (Eq. (2)): ⎧ ∂Mx (r, t) Mx (r, t) ⎪ ⎪ = −(ω0 (r) + γBd · ez )My (r, t) − ⎪ ⎪ ∂t T2∗ ⎪ ⎪ ⎪ ⎪ ⎪ Mz (t)Mx (r, t) ⎪ ⎪ − ⎪ ⎪ ⎪ M0 TRth ⎪ ⎪ ⎪ ⎨ ∂My (r, t) My (r, t) = (ω0 (r) + γBd · ez )Mx (r, t) − ∂t T2∗ ⎪ ⎪ ⎪ ⎪ ⎪ Mz (t)My (r, t) ⎪ ⎪ − ⎪ ⎪ ⎪ M0 TRth ⎪ ⎪ ⎪ ⎪ ⎪ Mx2 (t) + My2 (t) Mz (r, t) − M0 ∂Mz (r, t) ⎪ ⎪ − =− ⎩ ∂t T1 M0 TRth
(2)
assuming that the static magnetic field B0 lies along the z-axis. ω0 (r) is the local Larmor pulsation defined as ω0 (r) = −γB0 (r), γ being the nuclear gyromagnetic ratio of the xenon spins. Mx (t), My (t) and Mz (t) are the average magnetization components along the x-, y- and z-axes respectively, the averaging being performed over the volume effectively detected by the rf coil. T2∗ is the apparent transverse self-relaxation time and M0 is the thermal equilibrium magnetization. TRth is the radiation damping characteristic time for the magnetization at thermal equilibrium: 1 μ0 ηQ|γ|M0 . = (3) th 2 TR It depends on the filling (η) and quality (Q) factors of the probe. The inspection of system (2) reveals the existence of two non linearities: the first one through the radiation damping field and the second one through the dipolar field
Bd which induces that the spin dynamics of any magnetization voxel depends on all the others. These distant dipolar fields induce generally, for a spin system with a strong magnetization [26], nonlinear behaviors, such as dynamic instabilities [22,24] or the spectral clustering phenomenon [19,26]. The feature of the latter is the following: even if there is a distribution of B0 field along the sample, due to DDF, the magnetization is able to organize itself in different modes, precessing at well-defined resonance frequencies. Thus, instead of a broad resonance due to the field inhomogeneity, narrow lines are observed. 2.2 Analysis of the maser emission profile 2.2.1 Radiation damping equations Being dependent on the local variations of B0 via ω0 (r), the system (2) has no general analytic solution. A limit solution nevertheless exists. Firstly, if we assume that B0 is homogeneous over the whole sample of ellipsoid shape (ω0 (r) = ω0 ), results of the dipolar field theory indicate that Bd is constant over the sample, parallel to B0 and proportional to the dipolar field Bdip = μ0 Mz [26]. As already shown numerically, in these conditions during a maser emission, it only leads to a sweep of frequency which disappears when the modulus of the transverse magnetization is considered [27]. In the approach described below, we can consequently neglect its influence. These two restrictions let us deal with a unique homogeneous magnetization M (t) = M (r, t) ∀r, driven by the simplified equations written in equation (4) ⎧ ∂Mx (t) Mx (t) Mz (t)Mx (t) ⎪ ⎪ = −ω0 My (t) − − ⎪ ⎪ ∂t T2∗ M0 TRth ⎪ ⎪ ⎪ ⎪ ⎨ ∂My (t) My (t) Mz (t)My (t) = ω0 Mx (t) − − (4) ∂t T2∗ M0 TRth ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Mx2 (t) + My2 (t) ∂Mz (t) Mz (t) − M0 ⎪ ⎪ ⎩ =− − . ∂t T1 M0 TRth Assuming also that the nuclear spin has an infinite longitudinal self-relaxation time T1 , a maser emission can be analytically described by the Bloch equations given in equation (4), derived for spin dynamics of inverted thermal-equilibrium magnetization in the presence of radiation damping [14]. These equations can be modified for hyperpolarized spin systems. The general solutions for transverse magnetization M⊥ and longitudinal magnetization Mz are: M⊥ (t) = M0 TRth ρ sech ρ(t − t0 ) 1 th Mz (t) = M0 TR ρ tanh ρ(t − t0 ) − ∗ T2
(5) (6)
where sech x = 1/(cosh x), and t0 is the time where the maser signal is at maximum. ρ is a characteristic rate of the maser emission defined as: ρ=
|K| 1 − TRth T2∗
(7)
D.J.-Y. Marion et al.: Spectral and temporal features of multiple spontaneous NMR-maser emissions
where K is the enhancement factor of the polarization; at t = 0, Mz = KM0 and K < 0. As a consequence of equations 5 and 6, the spin dynamics of a NMR-maser emission can be characterized by the parameter ρ (Eq. (7)). Obviously, a better knowledge of the system can be obtained if the two parameters |K|/TRth and 1/T2∗ can be determined separately from experimental data. Finally a maser emission can be detected if ρ > 0 i.e. |K| 1 > ∗. th T TR 2 After an emission, the sign of the longitudinal magnetization can still be negative if 2 |K| > th ∗ T2 TR as inspection of equation (6) indicates and as experimentally observed [27]. 2.2.2 Model for data processing Extracting temporal and spectral features from the multiple maser emissions is not straightforward since, conversely to the theoretical model (Eq. (5)), the time dependence of the modulus of the transverse magnetization is not a single hyperbolic secant function [27]. In particular, frequency beats are present. They are tentatively explained as the superposed emissions of different modes resulting from DDF. Finally, during an emission the variation of the longitudinal magnetization induces a frequency sweep. Due to this last feature, it seems reasonable to disregard the absolute resonance frequency in a first approach and to consider the magnitude of the transverse magnetization. Nevertheless, because of the presence of frequency beats, it seems more suitable to work on the squared modulus to take benefit of the linearity of the Fourier transformation. Indeed, considering the superposition of two functions fa (t) exp(ıωa t) + fb (t) exp(ıωb t), fa,b (t) ∈ R, the square of its modulus: fa2 (t) + fb2 (t) + 2fa (t)fb (t) cos(ωa − ωb )t, allows the determination by Fourier transformation of the difference of their precession frequencies |ωa − ωb |. In the absence of a more general solution, which would be valid in the presence of DDF, it seems reasonable to try and describe the Fourier transform of this squared modulus as a superposition of Fourier transforms of the square of hyperbolic secant functions, since this last function is an analytical solution for a single maser emission (Eq. (5)). Considering the superposition of two maser emissions: M⊥ = Aeıωa t sech ρa (t − ta ) + Beıωb t sech ρb (t − tb ) (8) one finds:
359
The Fourier transform of the first two terms is: F (ω) = A2
πω πω ıωta ıωtb + B2 πω e πω e 2 sinh 2ρ 2 sinh 2ρb a
(10)
and thus the square of its modulus is:
2 πω +B |F (ω)| = A πω 2 sinh 2ρ b πω πω 2 2 2 + 2A B cos ω(ta − tb ) πω πω . (11) 2 sinh 2ρ 2 sinh 2ρ a b 2
4
πω πω 2 sinh 2ρ a
2
4
The first two terms of equation (11) are peak functions centered at zero frequency. Being both of the same shape and monotonous decreasing functions for ω > 0, they are expected to be hardly distinguishable by fitting procedures for ρa ∼ ρb [32]. This is confirmed by numerical simulations, which reveal that one can only determine an “average” ρ value, typically dependent on the line-width of the peak at zero frequency. The last term of equation (11) may lead to the apparition of waves on the Fourier transform pattern or may induce a narrowing of the main peak at zero frequency. In fact, because of the very fast decay of the function x2 / sinh2 x, the relevance of this modulation depends on the time difference between the two consecutive burst maxima, |ta − tb |, which should be longer than about 1/ρ. By considering separately the successive maser emissions with a confidence even larger than 98%, we have never found a case where adding this cos2 term does improve significantly the fit. This term has consequently been disregarded. The Fourier transform of the last term of equation (9) has no general analytical solution, except when ta = tb and ρa = ρb , where one has for the square of the modulus of its Fourier transform: A2 B 2
2
π(ω − (ωa − ωb )) a −ωb )) 2 sinh π(ω−(ω 2ρa
2
+A B
2
π(ω + (ωa − ωb ))
a −ωb )) 2 sinh( π(ω+(ω 2ρa
2 (12)
which is the same function as previously but centered at ±(ωa − ωb ). 2.3 Maser emissions and energy The energy lost by the nuclear spin system between the beginning and the end of a maser emission is simply: ΔEmagn = −Vs B0 (Mz (∞) − Mz (0)) = −Vs B0 M0 (Kf − Ki )
(13)
2
|M⊥ | = A2 sech2 ρa (t − ta ) + B 2 sech2 ρb (t − tb ) + 2AB cos(ωa − ωb )t sech ρa (t − ta ) sech ρb (t − tb ). (9)
where Vs is the sample volume, Ki and Kf are the magnetization enhancement factor at the beginning and at the
360
The European Physical Journal D
end, respectively. Assuming a correct description by the RD theory, one derives from equation (6): ΔEmagn = −2Vs B0 M0 TRthρ.
(14)
Equation (14) reveals a linear relation between the variation of stored magnetic energy inside the coil and the parameter ρ which defines the maser emission. On the other hand, the average power dissipated in the tuned circuit is [37,38]: P =
ωVs |2b1 |2 2μ0 Qη
(15)
where |2b1 | is the amplitude of the radiofrequency field created by the precessing magnetization. The calculation of the radiation damping time TRth allows its derivation: |2b1 | = μ0 ηQ |M⊥ | .
(16)
We can consequently determine the radiative energy emitted by the magnetization: ωVs 2 2 2 μ η Q |M⊥ (t)|2 dt. (17) ΔEcoil = P dt = 2μ0 Qη 0 Inserting the expression of the transverse magnetization in the presence of radiation damping (Eq. (5)), we find: ΔEcoil = 2Vs B0 M0 Trthρ.
(18)
ΔEmagn = −ΔEcoil .
(19)
Thus
We can hence conclude, that in the absence of longitudinal relaxation but in the presence of transverse relaxation, the whole nuclear magnetic energy present in the volume sample is transformed in radiative energy during a maser emission and received by the coil.
3 Materials and methods 3.1 Experiments The experimental protocol used to observe multiple spontaneous maser emissions was identical to that of experiments reported in reference [27]. Briefly, xenon was polarized by spin-exchange optical pumping [1] using our experimental apparatus [33]. The choice of the orientation of the quarter-wave plate was such that the final xenon spin temperature was negative. Laser-polarized xenon was separated from nitrogen buffer gas by condensation and transferred in the fringe field of the magnet to the NMR tube of interest, which contained degassed deuterated cyclohexane as solvent. Fast xenon dissolution and homogenized concentrations were achieved by shaking the tube vigorously. NMR experiments were performed at 293 K on a Bruker Avance II spectrometer with a magnetic field of 11.7 T equipped with a direct X-H Nalorac probe with
Table 1. Experimental conditions of the five experiments considered here. The labels 1 and 2 correspond to the quantities of xenon spins in the detection coil, which thus defines the radiation damping characteristic time. Quantitative values are deduced by multiplying the xenon concentration by the active volume of the NMR tube, about 350 μL for the 5 mm NMR tube (series L) and about 50 μL for the 3 mm NMR tube (series S). Name NMR tube diam. [Xe] (mol/L) Xe spin temp. number of masers
S1 3 0.18
