D F Escande

F Sattin

Breakdown of adiabatic invariance of fast ions in spherical tokamaks

The dynamics of fast particles corresponding to O(100) keV in NSTX, or of alpha particles in a properly rescaled reactor-grade spherical tokamak, is computed numerically through integration of the full equations of motion. The magnetic moment µ of these particles has large oscillations, and even chaotic ones, in a sizable domain of the machine. This has both practical and physical consequences. First, when µ has large oscillations, the use of guiding-center or gyrokinetic calculations for such orbits is questionable. Second, the capability of these particles to excite Alfvénic instabilities decreases, since the velocity of the particle fluctuates with respect to the phase-velocity of the Alfvén wave, which imposes a fluctuating sign to the energy exchanges with this wave. Even when chaotic orbits are present, the conservation of the toroidal momentum sets strong constraints about the volume available to the particles, and their radial diffusion stays bounded.

Introduction

Adiabatic invariants are fundamental tools in magnetized plasma theory. The conservation of the magnetic moment under slow variations of plasma parameters is a basic assumption of guiding-center theory and lies at the heart of gyrokinetic codes, which are presently the state-of-the-art instruments for the study of plasma microinstabilities. The existence of a sufficient number of constants of motion-including the adiabatic ones-ensures regularity of the motion; conversely, the lack thereof may-although not necessarily-imply sizeable chaotic motion. Adiabatic invariance is based upon the postulate of a wide separation between the particle Larmor radius ρ and cyclotron frequency Ω from the one side, and the corresponding length and frequency scales L, ω of the ambient magnetic field. In strongly magnetized laboratory plasmas this postulate is generally verified by a large margin. Yet, since ρ, Ω, L, ω are all dependent on local conditions, it may happen that, at selected locations in the particle phase space, the scale separation fails with potential consequences upon the conservation of the adiabatic invariants. Fast particles (beam ions or alpha particles) in spherical tokamaks (ST's) combine large Larmor radii and wide variations of the magnetic fields (small L's), and thus might experience breakdown of adiabaticity under realistic conditions. A few studies dedicated to this topic confirmed this feeling [1,2], and [3] showed that a part of fast particle orbits in MAST cannot be described by guiding-center theory. Section 3.1.1 of the review paper [4] states "the numerical study of the nonadiabaticity diffusion effect of fusion products seems to be very important for reactor scale ST". This paper investigates the nature of the dynamics of fast (beam-like) ions in a spherical tokamak, with an axis-symmetric NSTX geometry taken as a reference, or for alpha particles in a reactor-grade spherical tokamak with appropriately rescaled size and magnetic field. This is done through numerical integration of the collisionless complete equations of motion, not the guiding-center ones. The purpose of the paper is manifold: first, we wish to extend studies [1,2] by examining a much larger set of trajectories than presented in those papers. Second, we are interested in the distribution of the dynamics in adiabatic, superadiabatic and non-adiabatic. While genuine non-adiabatic dynamics may lead to chaotic motion and possibly to a strong increase of ion losses to the walls, by superadiabatic dynamics one refers to cases where the magnetic moment µ oscillates over time scales longer than the Larmor one around a constant mean value. As long as the time-averaged µ remains constant, dynamics is still regular in an axis-symmetric device; yet, finite excursions in µ may nonetheless produce some effects: for instance, increased pitch-angle diffusion due to Coulomb scattering and a decreased excitation of Alfvénic waves. It also invalidates guiding-center and gyrokinetic descriptions. Even remaining in the collisionless regime, we will show that the boundary between superadiabatic and non-adiabatic motion is fuzzy: over very long times if compared to the bounce time, but still below the experimental collision frequency, a transition from a seemingly former one to the latter may take place.

Mathematical tools

We use a magnetic geometry corresponding to an axis-symmetric NSTX spherical tokamak, implementing the recipe suggested by Yavorskij et al [5]. Since we aim at reproducing a realistic ST equilibrium (and specifically one close to NSTX), but not necessarily a specific experimental one, we choose the numerical values of the parameters within the ranges suggested in [5], without caring for picking up the exact combination required for getting one of the equilibria published therein. The (R, Z, φ) set of cylindrical coordinates is related to the r flux surface radius and χ poloidal angle through (29,38,39) in [5] (see also its figure 1 and related discussion):

R = R c + ∆(r) + r cos(χ) (1)
Z = k r sin(χ)(1 -Λ(r) cos(χ)) α (2)
In ( 2), the Shafranov shift ∆, the elongation k, the triangularity Λ have been taken by the curves published in figure (5) of [5], corresponding to the NSTX case with β = 23%. The other parameters are R c = 1.25 a (a = minor radius), α = 1 [6]. No up-down asymmetry was included. The magnetic fields are defined as in (3-7) of [5]). Their definition involves the safety factor q(r), the poloidal current J(r) and the toroidal current I(r), which have been taken from the curves plotted in figures (13,15) of [5].

A problem of self-consistency arises, since we use in the calculation of magnetic fields the quantities I, J, q as given known functions, whereas their argument is the flux coordinates r, which should be self-consistently related to the magnetic field themselves. There is no warranty, therefore, that the computed fields are truly magnetic fields, i.e., divergencefree. In order to cure this defect, we clean the fields, by removing a quantity

B → B -∇σ (3)
with σ defined by Poisson equation

∇ 2 σ = ∇ • B (4)
The correction, which turns out to be fairly small, ensures B to be a valid magnetic field. Finally, we write the magnetic potential A = {0, A Z (R, Z), A φ (R, Z)}; its components are defined as

A Z = R dR B φ (R , Z) (5)
∂ 2 A φ ∂R 2 + ∂ 2 A φ ∂Z 2 = -(∇ × B) φ (6)
The latter equation is solved numerically. Figure (1) displays the contour plot of the local Larmor period 2π/B: besides being straightforwardly related to B, plotting this quantity is useful since it yields a visual insight of the characteristic time scale involved, which will be useful in the following. Particle trajectories are computed from the canonical equations of motion with Hamiltonian

H = P 2 R 2 + (P Z -A Z ) 2 2 + (P φ -A φ ) 2 2 . ( 7
)
Normalized units are used throughout the whole text: particle charge and mass are taken unitary; length is normalized to the torus minor radius a, and the magnetic field to a reference field B 0 . In order to convert from normalized units, it is convenient to use Larmor radius as a gauge: with values a = 0.7 m, B 0 = 0.3 T, a 100-keV deuteron with a velocity perpendicular to the magnetic field has a normalized Larmor radius ρ/a ≈ 0.2. In order to compute particle trajectories reliably, we have implemented a symplectic partitioned Runge-Kutta algorithm of sixth order, as described in [7]. The time step was always checked to stay well below 1/1000 of Larmor period, energy was conserved in all simulations with a relative precision not worse than 10 -6 and usually better by orders of magnitude.

Since the magnetic topology is held fixed, the only control knobs in our simulations are the particles' initial conditions. There are five of them, since the angle φ is an ignorable coordinate. In the following simulations we will check the effect of varying them all, but following a few guidelines. We use R as the main control parameter among the spatial coordinates; all particles are initialized on the equatorial plane Z = 0. Concerning the velocity degrees of freedom, we consider mostly a scenario of "fast particles", whose typical Larmor radius is of the order of one tenth of the minor radius, like those considered in the earlier study [1]. Larger Larmor radii would be unrealistic in a device; employing smaller Larmor radii, conversely, non-adiabatic features are expected to be more and more difficult to detect. We will briefly return to this at the end of section 3. Notice that in higher-magnetic-fields concepts of spherical tokamaks, such as those outlined in [8], this class of particles would refer to fusion-produced alpha ions, not to beam ions.

In an axis-symmetric geometry, two exact constants of motion exist: the total energy and the momentum along the ignorable coordinate φ. The existence of a third constant of motion ensures regularity of the motion but, conversely, the lack thereof does not necessarily imply sizeable chaotic motion. In order to confirm the existence of chaos we relied upon Poincaré maps, produced by recording particles' intersections with the plane Z = 0 and dZ/dt > 0. In a few cases we validated our conclusions with the explicit check of the existence of exponential separation between nearby trajectories. We finish this section by pointing out that our simulations are collisionless, thus are meaningful as long as they do not exceed typical collision times. We estimated that the slowing-down time for a fast ion in a NSTX plasma is about 1 ms. For comparison, our longest simulations, used for producing Poincaré maps, run up to a fraction of ms when converted into physical units, hence we feel confident that collisional effects do not alter our conclusions.

Results

Numerical findings: a selection

In the following figures we provide a representative selection out of our whole set of simulations. In this first set, all particles are initialized with dimensionless kinetic energy = 4.5 × 10 -4 ; then, the Larmor radius, averaged over the interval 0 ≤ t ≤ 4000, ranges roughly between 0.75×10 -1 and 1.5×10 -1 , as anticipated in the previous section. Since the same numerical trajectories will be presented in different plots, in order to help the reader avoiding confusion, each individual simulation will be uniquely tagged by the same color and progressive number, throughout the text. The joint conservation of the kinetic energy K and of the toroidal momentum P φ sets stringent constraints about the volume available to the particles. To start with, we write

v φ = p φ -A φ (R, Z) (8)
v 2 R + v 2 Z = 2K -v 2 φ = 2K -(p φ -A φ ) 2 (9)
For the chosen conditions, the initial v φ is of order 10 -2 or lesser, whereas A φ may vary within a range of about 10 -1 (see figure 2), and 2K = 9 × 10 -4 . Hence, the right hand side of the last equation is not warranted to be positive everywhere: the particle is constrained within subregions of the whole NSTX volume. In figure (3) we show the allowed regions (R, µ) on the equatorial plane for different values of P φ . It is apparent how the actual allowed region is just a tiny fraction of the whole potential volume. Indeed, the range available to µ is still smaller. We rewrite, on the basis of the second of (9

) v R = w cos(θ), v Z = w sin(θ), w 2 = 2K -v 2 φ
, and θ an appropriate angle. Through simple algebra it is possible to show that µ can be written as

µ = K B - B φ (p φ -A φ ) + 2K -(p φ -A φ ) 2 (B R cos(θ) + B Z sin(θ)) 2 2B 2 (10)
The first term, K/B is just the maximum reachable µ, when the whole kinetic energy is in the perpendicular component of the velocity. While, a priori, µ may vary over the whole range (0, K/B), the previous equation shows that the dynamically allowed range of µ is actually that obtained by varying θ in the interval (0, 2π). Finally, note again the presence of the term 2K -(p φ -A φ ) 2 inside the square root which has the same meaning as in Eq. ( 9) of delimiting the region of real space accessible to the particle. Figure (4) summarizes our main findings about the conservation of µ. It shows the pairs (R, µ) in correspondence of the intersection of the trajectory with the plane Z = 0. Strictly speaking, they are not Poincaré sections, since these should involve phase-space coordinate as, e.g., (R, P R ). The chosen mapping is not area-preserving, but retains the other interesting properties of Poincaré maps, i.e. KAM tori correspond to continuous curves and chaotic zones to sets of scattered points. In such a representation, if µ is conserved, each orbit appears as a series of horizontal segments. From now on, therefore, we will not distinguish between the two sections and refer to these mappings as Poincaré maps. Each row of figure (4) corresponds to a given value of P φ . In order to get a feeling about the scattering of the points throughout the dynamically available region of the parameters space, we shade the forbidden region, according to the constraints set by Eqns. [START_REF] Garren | Proceedings of the Second UN International Conference on the Peaceful uses of Atomic Energy[END_REF][START_REF] Hastie | Proc. 3rd Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research[END_REF]. Besides this, we plot the curves corresponding to the maximum µ reachable at each spatial position (R, Z = 0) : µ max = K/B. We show several orbits, each corresponding to different initial conditions. The most striking feature of these plots is the fact that a large fraction of these orbits have large fluctuations of µ. In the first row, corresponding to the more external considered orbits, the relative variation of µ is between 30 and 160%. In the second and third row, similar orbits are present, but the most central orbits with µ ≤ 1 have a very good conservation of µ. The conservation of P φ strongly limits the radial excursion of orbits, even chaotic ones. So, the strong non conservation of µ does not mean a strong loss of confinement.

We identify three sets of orbits. A first one corresponds to chaotic orbits, since their points do not match with a continuous curve, but fill a finite area. Such orbits are present for each of the three smaller values of P φ . µ is clearly not constant, since it spans a large part of the [0, µ max ] domain. A second set of orbits displays a small scatter of µ. They are regular, since their points define a continuous curve (orbits labelled # 42,10,11,18). There is, finally a third type of orbit which features a significant dispersion in µ, yet the overall appearance of the trajectory is regular (# 17,19). The second and third regimes represent a transition between the truly adiabatic and the super-adiabatic dynamics described by [1]: µ oscillates over a time scale substantially larger than the Larmor period, but does not drift away from a constant average value. The corresponding dynamics is still regular. Instead, the first set of orbits is a genuine example of non-adiabatic/chaotic dynamics.

Analysis of the non-adiabatic dynamics

We now investigate the physical causes which lead to the genuine non-adiabatic chaotic dynamics and to the failure of the guiding-center approximations. The magnetic moment is associated to the Larmor rotation around the magnetic field axis which, in a uniform magnetic field, corresponds trivially to a closed circuit. In a curved magnetic field, instead, no such circuit can be drawn any longer, but the difference in the position after a whole Larmor rotation, is smaller and smaller as particle velocity decreases. This difference is by no means small at the positions where µ changes appreciably. While performing its Larmor rotation, the particle simultaneously drifts. In a static magnetic field, the well-known gradient and curvature drifts exist:

u ∇B = K ⊥ B × ∇B B 3 , u curv = 2K R curv × B R 2 curv B 2 (11)
Within a Larmor period, which-in our units-is T = 2π/B, particles drift by a length which, on the basis of the structure of the previous equation, scales like KB -2 . This has to be compared with the movement due to Larmor rotation, i.e., the Larmor radius, which scales like K 1/2 B -1 .Therefore, it is intuitive that the smaller the magnetic field becomes locally, the more difficult it is for the circuit to close. In order to make quantitative our statement we resort to two numerical markers. They are expected to convey roughly the same information, yet their sensitivity may be different. The first quantity is

c(t) = μ(t) • b(t) (12
) b is the unit vector of the local magnetic field; μ is the unit vector defining the local direction of the magnetic dipole moment made by the microscopic spire produced by the rotating particle (we describe in the Appendix A, for the reader's convenience, how μ is computed from numerical data). Thus, c, the cosine of the angle between the two arrays, quantifies the degree of alignment of μ with the direction of the local magnetic field. In an adiabatic regime, μ remains aligned with b at all times, and c = 1 everywhere.

The second marker quantifies the amount of non-uniformity of the magnetic field orientation as seen by the particle during one rotation:

n(t) = b(t) • b(t + T ), T = 2π/B (13)
For adiabaticity to hold, n must stay close to 1. Notice that both markers are meaningful for curved magnetic fields: in straight magnetic fields, both c(t) and n(t) are always unitary, even though non-adiabatic dynamics may exist. Figure 5 shows time traces for several quantities: the averaged moment µ (where the fast oscillations over the Larmor time scale are removed through a moving average µ done over a time window equal to the local Larmor period 2π/B, in order not to overburden the plot), the R and Z coordinates, the c and n markers. The plots refer to the same runs as figure (4), however the visualized time span is much shorter than that employed for producing the Poincaré maps: about 1/15th. In order to appreciate the time scales involved, we remind that, with the chosen units, the Larmor period is in the range 20 ÷ 40 time units (see figure (1). We enumerate the conclusions resulting from the inspection of the plots: (i) Superadiabatic dynamics corresponds to slow large oscillations of µ over a mean value which remains constant in time. Conversely, genuine non-adiabatic dynamics features intervals where µ is fairly constant separated by sudden jumps (i.e. occurring over time scales below or about the Larmor period) towards a fairly different value. These results are consistent with earlier studies [1,2,[START_REF] Garren | Proceedings of the Second UN International Conference on the Peaceful uses of Atomic Energy[END_REF][START_REF] Hastie | Proc. 3rd Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research[END_REF][START_REF] Howard | [END_REF]. (ii) Perfect alignment is seldom reached, c < 1 almost everywhere, but c has severe drops in coincidence with µ jumps. (iii) The condition n = 1 is well fulfilled everywhere except, again, in coincidence with jumps of µ . Thus, both c and n are good proxies for the breakup of the adiabaticity condition. The breakup of adiabatic dynamics is here to be understood as a consequence of an orbit crossing a region of (a) small magnetic field amplitude and (b) large variation of the orientation of the magnetic field, with (c) a large enough velocity. Conditions (a) makes large the argument T in Eq. ( 13) and, together with conditions (b,c), entails a large difference between b(t) and b(t + T ) such that in the particle frame the condition n ≈ 1 does not hold. Equivalently, during one single Larmor rotation, the magnetic field rotates too fast for the magnetic moment to align with it. We expect the departure from adiabatic dynamics to be smaller and smaller as the particle velocity decreases. Accordingly, we made a few exercises reducing it. Figure ( 6) displays a sample of the results, and shows the non-conservation of µ for particles with reduced speed, confirming our expectations. It is interesting to note that the smallest velocity appearing in the simulation is in the thermal velocity regime and still features a faint degree of non-adiabaticity. This result must not be overestimated, though: at these small energies, collisional effects interfere already at these time scales.

We address now the issue of the boundary between superadiabaticity and nonadiabaticity/chaos. Figure (7) shows the time traces of µ corresponding to trajectory #42 over the whole time range used for producing its Poincaré map, splitted into four subintervals. From figure (4) one expects this orbit to be non-adiabatic, yet from figure Figure 6. Time traces of the scaled ∆ < µ > quantity. All particles are initialized at R = 0.5, Z = 0 with the same φ-velocity -6.7 × 10 -4 , and thus the same momentum P φ = -0.5. Total kinetic energy is varied, though: K = 4.5 × 10 -4 (green curve); K = 1.2×10 -4 (orange curve); K = 7.25×10 -7 (blue curve). Because of the extremely small velocity in this latter case, the simulation has been run for twice the standard time span, in order to allow for non-adiabatic effects to appear. Actually, they start showing exactly when the previous two simulations stop. (5) its time trace appears closer to superadiabatic. Actually, one recovers that the time trace is very close to superadiabatic over a very long interval of time, up to roughly t = 15000 but, eventually it turns into a non-adiabatic one. Thus, the boundary between the different kind of dynamics does not look sharp, but rather fuzzy.

A final comment concerns the issue of the isotropy of the particle distribution in phase space. Our ambition was to treat on the same footing beam ions and alpha particles. Yet, the former ones have a highly anisotropic distribution in phase space, though, unlike the latter ones, and one may wonder whether and to which extent anisotropy affects the breakdown of the adiabatic invariance. This question is partially and implicitly answered by the analysis of the behavior of the trajectories as a function of the toroidal momentum (Figs. 3,4 and related discussion). Indeed, since the total energy of the particles is held fixed, varying the toroidal momentum-at fixed spatial location-is equivalent to varying the pitch.

Discussion and conclusion

We studied the dynamics of fast beam-like particles in an axis-symmetric version of NSTX, or of alpha particles in an appropriately rescaled reactor-grade device. Our results show that the magnetic moment µ of these particles have large oscillations, and even chaotic ones, in a sizable domain of the particle initial conditions. The breakup of adiabaticity occurs when an orbit with a large enough velocity crosses a region of low magnetic field amplitude with a large variation of the orientation of this field. These large oscillations of µ have both practical and physical consequences. First, the use of guiding-center or gyrokinetic calculations for such orbits is questionable. Second, the capability of these particle to excite Alfvénic instabilities decreases, since the velocity of the particle fluctuates with respect to the phase-velocity of the Alfvén wave, which imposes a fluctuating sign to the energy exchanges with this wave. Indeed, Section IV of the tutorial paper about mechanisms of energetic-particle transport in magnetically confined plasmas [13] shows that the essence of Alfvén wave excitation by fast particles is related to wave-particle resonance. Such a resonance corresponds to an exact linear relation between the wave frequency and the orbital frequencies of the particle. For orbits like some of those in figure 5 where µ is not conserved on a bouncing period in Z, these orbital frequencies cannot be defined, which rules out wave-particle resonance. These two facts might explain a result displayed in figure 8 of [12], which shows the total neutron rate drop in NSTX versus scaled TAE amplitude: the numerical calculation performed with the guiding-center ORBIT code overestimates the experimental value of neutron rate drop due to the waves by a factor 2 when using the measured TAE mode amplitude. The wave amplitude must be scaled down by a 0.7 factor to get agreement with the experimental value. Our calculations imply that Alfvén waves are inefficient in the outer part of the confined plasma, which are good news for the confinement of fast particles.

Even when chaotic orbits are present, the conservation of the toroidal momentum P φ sets strong constraints about the volume available to the particles, and their radial diffusion stays bounded. However this will no longer be the case in the presence of magnetic ripple.

Figure 1 .

 1 Figure 1. Contour plot of the local Larmor period 2π/B.

Figure 2 .

 2 Figure 2. Left plot, radial profile along the equatorial plane Z = 0 of the toroidal component of magnetic vector A φ ; right plot, Z-component of the magnetic vector A Z .

Figure 3 .

 3 Figure 3. Radial plot along the equatorial plane Z = 0 of the square root of Eq. (9) for different values of P φ . In all cases 2K = 9 × 10 -4 .

Figure 4 .

 4 Figure 4. Each subplot includes Poincaré maps corresponding to two or more trajectories, in the plane (R, µ). Each trajectory is identified consistently throughout the whole paper by a unique color and identifier number. Plots are grouped on the basis of the toroidal momentum P φ : from top to bottom P φ = -0.075, -0.09, -0.1, -0.125. Dashed areas mark the regions forbidden to the particle according to the discussion around (9, 10).

Figure 5 .

 5 Figure 5. All plots show time traces of µ (blue curve), R (orange curve), Z (green curve), c (red curve), n-1 (black curve) for a few trajectories, both chaotic and regular. The time window extends up to t = 4000: about 1/15th of the whole simulation time used for producing Poincaré maps.

Figure 7 .

 7 Figure 7. Time traces of µ splitted into four subintervals, corresponding to orbit #42

Acknowledgments

The authors wish to thank two anonymous referees for careful reading of the manuscript Appendix A

Vector μ may be computed using just the recorded particle trajectory s(t) in terms of the Serret-Frńet triad: the tangent to the trajectory T, its normal N, and the binormal

and

This procedure, however, involves computing second derivatives of s(t) which may be very noisy for discretely sampled data. For numerical purposes, thus, we adopted the following procedure: first, the velocity of the particle is computed from the recorded trajectory

The acceleration a may be written both in terms of the derivative of v (recalling that dv/dt = 0)

as well as in terms of the law of motion:

The vector N is parallel to the normal N, but is not unitary. By equating these two expressions we get

This latter recipe involves only s and its first derivative, and is numerically much less noisy.


L Chouhan

G Bouzerar

S K Srivastava
email: sk.srivastava@cit.ac.in

d 0 Ferromagnetism in Li-doped ZnO Compounds

Keywords: ZnO, Li-doping, d 0 ferromagnetism, Spin Glass Phase

Recently, d 0 ferromagnetic materials have been projected as one of the promising novel materials for spintronics applications. In this work, we have studied Li-doped ZnO compounds, i.e. Zn1-xLixO (x=0, 0.02, 0.04, and 0.06) samples, prepared by the solid-state reaction route method. From the study of crystal structure using X-ray diffraction (XRD) patterns, it is evident that the prepared materials have been formed in a single-phase of the hexagonal wurtzite structure. The refinement of the XRD patterns suggests that there are very small changes in the lattice parameters upon Li-incorporation in ZnO. The average crystallite size (SC), estimated from XRD patterns was found to be in the range of 35-50 nm. The microstructural study by scanning electron microscope reveals the uniform morphology of the grains of the order of 50-70 nm. The energy dispersive spectrum indicates that no unwanted ferromagnetic impurities have crept into the final prepared samples. The measurement of the temperature (T) variation of magnetization (M) with SQUID magnetometer indicates that undoped ZnO exhibits diamagnetic property but all Li-doped compounds exhibit room-temperature ferromagnetism and with a magnetic irreversibility behavior between zero-field cooled and field cooled M-T data. From the magnetization versus field measurements at 3 and 300 K, it is observed that Lidoped samples exhibit ferromagnetic loops with ultra-soft coercivity (~50 Oe) and with a maximum saturation magnetization of 0.10 emu/gm for x= 0.02 sample, which decreases with the increase in Li concentration.

Introduction:

Recently, doping with non-magnetic elements (e.g. Li, Na, K, Mg, Ag, etc.) in the nonmagnetic host oxides, such as ZnO, TiO2, SnO2, ZrO2 etc. has gained a lot of attention because of their possible application in spintronics [1]. In particular, it was emphasized that ferromagnetism can be observed in such non-magnetic element doped oxide materials and it was referred to as d 0 or intrinsic ferromagnetism. There exist a variety of calculations that predicts ferromagnetism even in the un-doped oxide and the observed d 0 magnetism was credited to originate from oxygen vacancies/defects [1]. To overcome the difficulties of controlling intrinsic defects, one model was proposed by Bouzerar et al., where it was proposed that nonmagnetic element doping in oxides such as HfO2 or ZrO2 could lead to ferromagnetism (FM) with high Curie temperature [2]. It is important to mention that in the previous past three decades, ferromagnetic ordering in transition metal (TM) doped oxide materials were reported extensively and it was well documented [3][4][5][6][7][8][9][10]. But there is a debate about whether the observed magnetism is an extrinsic effect arising from impurity phases, or it is an intrinsic property [11]. Thus, recently d 0 ferromagnetic materials were projected as an alternative pathway to TM doped oxide materials and expected to lead clean materials for spintronics applications. Indeed, many ab-initio studies have predicted d 0 ferromagnetism in oxides, such as SnO2, TiO2, ZrO2 doped with different non-magnetic elements [12][13][14][15][16][17][18][19][20][21][22][23] and it was observed experimentally as well [24][25][26][27][28][29][30][31][32][33]. Recently, our group has observed d 0 ferromagnetism in Kdoped SnO2 [27], K-doped rutile TiO2 [28], Li-doped SnO2 [29] and Ag-doped monoclinic ZrO2 compounds [30].

Zinc Oxide (ZnO) has long been used for many applications such as piezoelectric transducers, varistors, and transparent conducting films, etc. Interestingly, d 0 ferromagnetism was observed in undoped ZnO thin films also, wherein a successive transition from ferromagnetism to paramagnetism and diamagnetism as a function of film thickness was observed [34][35][36][37]. The ferromagnetic order in ZnO thin film was believed to be originated from defects induced [34][35][36][37]. However, the ZnO compound prepared in bulk form was found to exhibit diamagnetic behavior [38][39]. In addition to un-doped ZnO compound, the possibility of d 0 ferromagnetism in ZnO was explored by doping with various non-magnetic elements, such as Cu-doped ZnO [40][41], alkali metal-doped ZnO [42], Li-doped-ZnO [39,[43][44][45][46][47], Mgdoped ZnO [48] and Ag-doped ZnO [49]. Particularly, the magnetic properties of Li-doped ZnO [39,[43][44][45][46][47] were studied by a few research groups by preparing them in different forms of materials using various methods. Lin et al. [43] reported that Li-doped ZnO thin films exhibit paramagnetic behavior; whereas, ferromagnetic behavior was observed in Li-doped ZnO, prepared in the form of nanorods [39], thin-film [45], and nanoparticles using the sol-gel technique [46] and using chemical precipitation method [47]. Chawla et al. [39] observed that Li-doped ZnO nanorods exhibit d 0 ferromagnetism with high Curie temperature and it was proposed that the substitutional Li dopant induces magnetic moments on neighboring oxygen atoms. Whereas, few other groups [45][46] pointed out that complex defects are responsible for controlling and tuning the observed d 0 ferromagnetism in Li-doped ZnO compounds. The maximum saturation magnetization in these compounds was observed to be 0.055 emu/gm. Thus, Li-doped ZnO compounds have led to different results on magnetism and a strong dependency of magnetic property on the method used to prepare and form of the materials was observed.

In a quest of studying the magnetic property of Li-doped ZnO compounds, we have synthesized these compounds in bulk form at equilibrium conditions using the solid-state route method. The Li-doped ZnO bulk compounds, prepared by solid-state route method have the obvious advantage of eliminating the possibility of clustering/defects which is a serious drawback in the materials prepared in thin-film form or prepared by chemical route. We have demonstrated that these Li-doped compounds exhibit room-temperature ferromagnetic behavior.

Experimental Details

The polycrystalline samples of Zn1-xLixO (x=0, 0.02, 0.04, and 0.06) were prepared by solid-state reaction route, using high-purity starting materials of ZnO (Sigma Aldrich, purity-99.999% trace metals basis) and Li2CO3 (Sigma Aldrich, purity-99.997 % trace metals basis).

The maximum amount of any kind of trace magnetic impurities (such as Fe and Co) in the starting materials was found to be less than 2 ppm (parts per million) as mentioned in the supplier chemical analyses report. The required amount of each starting compound (ZnO and Li2CO3) was calculated in the stoichiometric ratio and the weighing was carried out using an electronic balance. These starting compounds were grinded using a mortar and pestle under acetone for a proper mixing and for good homogeneity of the sample. After grinding the starting materials, the pre-sintering of the samples in powder form was carried out in air at 100 0 C, 200 0 C, and 300˚C for 10 hours at each temperature. The final annealing of these samples in pallet form was done in the air at 500 0 C for 20 hours and these samples were used for the study of crystal structure and magnetic properties. The crystal structure and phase purity of these prepared samples were studied by recording powder X-Ray diffraction (XRD) patterns at room temperature using a Philips diffractometer operated in θ-2θ Bragg-Brentano goniometer geometry and by employing CuKα (1.541874 Å) incident radiation beam. Data collections were carried out over a scan range of 2=20 to angle 2=90 with a step of 0.02. Microstructural study was carried out by recording microstructural images using a ZEISS-ultra-plus scanning electron microscope (SEM). The elemental analysis was done with BRUKER energy dispersive spectrometer (EDS) attached in-situ with SEM. The measurements of the temperature variation of magnetization (M-T) under zero-field cooled (ZFC) and field cooled (FC) conditions were carried out under an applied field of 500 Oe and for a temperature range of 3-300 K using commercial SQUID magnetometer (Quantum Design MPMS XL). The field dependence of magnetization (M-H) curves, were measured in the field range of ± 5 T at 3 and 300 K. The magnetic measurements were performed with utmost care and, handling of the samples was done with plastic tweezers.

Results and Discussion:

To understand the crystal structure and phase purity of all prepared Li-doped ZnO samples, X-Ray diffraction (XRD) patterns were recorded at room temperature and it is presented as Figure 1. Undoped ZnO and Li-doped ZnO samples were found to crystallize in single-phase form with hexagonal ZnO wurtzite structure (space group P63/mc). The XRD data indicates that no secondary phase is present in the compounds within the instrumental limit of the X-ray diffractometer. In order to further ascertain the phase purity and to estimate the lattice parameters 'a' 'b', 'c' and cell volume (V), the recorded XRD patterns were refined with the help of the Fullprof program software by employing the Rietveld refinement technique [START_REF] Young | The Rietveld Method (International Union of Crystallography)[END_REF].

The background coefficient parameters, peak shape parameters (α, β, γ, u, v, w), and lattice parameters (a, b, c) were refined during Rietveld refinement. The XRD patterns for all samples could be refined by using the P63/mc space group. The XRD patterns along with Rietveld refinement for ZnO and 6% Li-doped ZnO samples are shown in Figure 2 It is seen that the experimental XRD data matches perfectly with the Rietveld software calculated XRD data. The refinement of the XRD data indicates that the lattice parameters for the un-doped ZnO compound are found to be a=b= 3.2508 Å, c= 5.2076 Å, and V= 47.66 Å 3 , and these values are comparable with those reported by other groups [40]. The incorporation of Li atoms into ZnO lattice leads to small change in the values of lattice parameters (only in the fourth decimal place), as demonstrated in Figure 3. This can be understood in terms of comparable ionic radii of Zn ions (with ionic radii of 0.74 Å) and Li-ions (with ionic radii of 0.60 Å). However, it is very difficult to comment about whether Li-ions are located at the substitutional sites or interstitial sites. The crystallite size (SC) was calculated from the FWHM (full width at half maximum) data obtained from the refinement and using the Debye Scherrer formula. The crystallite size of un-doped ZnO is estimated to be 35 nm, whereas SC for 2, 4, and 6 % Li-doped ZnO samples are found to be 37, 49, 38 nm respectively. Followed by the study of crystal structure and micro-structure, the magnetic properties of these prepared samples were studied. As a first step, the magnetic properties of all the starting compounds viz. ZnO and Li2CO3 have been checked and they clearly exhibit diamagnetic behavior (not shown). The measurements of magnetization versus temperature (M-T) and; field variation of magnetization (M-H) curves (measured at 3 and 300 K) of un-doped ZnO compound indicate a clear diamagnetic behavior, as demonstrated in Figure 5. Similar diamagnetic behavior in the ZnO compound was reported by other groups also [38][39]; but this is unlike other reports on ZnO thin film, where ferromagnetism was observed for the undoped ZnO compound due to defects [34][35][36][37]. The measurement of zero-field cooled (ZFC) M-T curves (Figure 6) for all Li-doped ZnO compounds indicate that there is a very slow dependency of magnetization with temperature throughout the measured temperature range and there was no ferromagnetic transition upto 300 K, indicating that the transition temperature of all Li-doped ZnO compounds is well above the room temperature. Moreover, an upturn/peak in ZFC M-T curves was observed for 2 and 4 % Li-doped ZnO samples at 65 K and 55 K respectively. Here, the characteristic temperature was estimated from fitting the peak with a Gaussian function. We did not observe this feature for 6% Li-doped ZnO sample. This observed feature could originate from small antiferromagnetic phase embedded in the dominant ferromagnetic matrix or due to some disorder in the lattice. The measurement of field cooled (FC) M-T data (as shown in Figure 6) indicates a magnetic irreversibility behavior between ZFC and FC curves for 2, 4 and 6% Li-doped ZnO compounds. The magnetic irreversibility behavior of these samples starts out from 300 K itself and a continuous increase in the value of FC magnetization on lowering the temperature is observed. Moreover, a prominent drastic upturn in the FC magnetization curve, below the characteristic temperature is observed for 2 and 4% Li-doped ZnO samples. Such magnetic irreversibility between ZFC and FC M-T curves may be attributed to the presence of magnetic spin glass phase, possibly arising due to the competing interactions or due to the presence of some disorder in the lattice.

A similar kind behavior i.e. presence of magnetic spin glass phase was observed in Li-doped ZnO nanoparticles prepared by chemical sol-gel technique [46]. To get a further understanding of the magnetic properties of these samples, we have measured field dependence of magnetization in the field range of ± 5 T at 3 K and 300 K for all Li-doped ZnO compounds and they are presented in Figure 7. The M-H curves measured at 3 and 300 K show typical ferromagnetic hysteresis loop for all Li-doped ZnO compounds. The M-H curves measured at 300 K exhibit ultra-soft coercivity, whereas the M-H curves measured at 3 K exhibit enhanced coercivity (as shown in the inset of Figure 7b). The coercivity was found to be for 50, 25 and 48 Oe for 2, 4, and 6% Li-doped samples respectively. These observations indicate that all Li-doped compounds exhibit soft ferromagnetic properties. The value of saturation magnetization (MS) at 300 K (as shown in Figure 8), is observed to be 0.025, 0.008, and 0.002 emu/gm for 2, 4, and 6% Li-doped samples respectively, which is found to be enhanced at 3 K. The value of MS at 3 K is found to be 0.098, 0.019, and 0.004 emu/gm for 2, 4, and 6% Li-doped samples respectively. Thus, at both the measured temperatures, the value of Ms decreases with the increases of Li concentration. Chawla et al. [39] reported the saturation magnetization value of 0.055 emu/gm for 2% Li-doped ZnO, which was the largest MS value, attributed to substitutional Li. However, Awan et al. [41] have demonstrated that the saturation magnetization of 0.055 emu/gm is due to interstitial lithium. Our measured values appear to be higher in comparison to all previously reported values. From the crystal structure study, it is observed that the prepared compounds have been formed in pure single-phase and the incorporation of Li atoms into ZnO lattice leads to very small changes in the lattice parameters (only in the fourth decimal place). The magnetic properties, measured from M-T and M-H curves indicate that undoped ZnO compound exhibit diamagnetic behavior, whereas the incorporation of Li into ZnO lattice provoked ferromagnetic behavior with a transition temperature much beyond room temperature. The value of saturation magnetization was found to decreases with the increase in Li concentration. It should be noted that we have not observed any secondary phase from the crystal structure and thus the observed d 0 ferromagnetism is intrinsic in nature. In the recent theoretical model of Ref. [2], it was suggested that three physical parameters are essential to explain induced d 0 ferromagnetism in the case of a direct non-magnetic cationic substitution in oxide materials: (i) the position of the vacancy (defect) induced impurity level (ii) the density of carrier per defect/vancancy and (ii) the electron-electron interaction strength. It was pointed out that the optimum window of the density of carrier is very important to tune room-temperature ferromagnetism in the nonmagnetic element (e.g. Li, Na, K, etc) substitution induced magnetism in oxides. Beyond the optimum window, the magnetic ordering vanishes as magnetic couplings are destroyed by Ruderman-Kittel-Kasya-Yashida (RKKY) type oscillations or antiferromagnetic superexchange. It was further emphasized by this model that when the on-site potential controlling the impurity level's position is too small, antiferromagnetic nearest-neighbor couplings exist, whereas the coupling become ferromagnetic with the increase of the on-site potential strength.

However, with a further increase of this potential, some couplings become antiferromagnetic

Conclusion

To conclude, we have studied the crystal structure and magnetic properties of Li-doped

  . Here the experimental data are shown as open circles and the calculated intensities are shown as solid lines. The bottom line represents the difference between measured and calculated intensities.

Figure 1 :

 1 Figure 1: X-ray Diffraction patterns of Zn1-xLixO (x=0, 0.02, 0.04, and 0.06) compounds.

Figure 2 :

 2 Figure 2: XRD patterns along with Rietveld refinement for (a) ZnO (b) 6% Li-doped ZnO samples. The circles represent experimental points and solid line represents Rietveld refined data. The dotted lines show the difference between experimental and refined data.

Figure 3 :

 3 Figure 3: Variation of crystal structure parameters a, c, and V, along-with error bar of Zn1-xLixO (x=0, 0.02, 0.04 and 0.06) compounds.

  Figure 4 (a) and (b) for 2 and 4% Li-doped ZnO samples respectively. The grain size was determined from the analyses of the grain size distribution with software integrated with SEM and; by fitting the distribution of grain size with a Gaussian function. One representative fitting is shown for 4% Li-doped ZnO sample in Figure 4 (c) and the grain size is found of the order of 50-70 nm. In order to ascertain if any other unwanted ferromagnetic impurities (such as Fe or Co etc.) have not crept into the final prepared samples, the elemental analysis of the samples has been carried out by recording EDS spectra. One typical EDS spectrum of 4% Li-doped ZnO sample is shown in Figure 4 (d). Since Li is very light element, it was beyond the instrumental limit of the EDS to detect the distribution of Li. The position for magnetic elements such as Fe and Co is marked as a vertical line, indicating the absence of any magnetic impurity in the sample.

Figure 4 :

 4 Figure 4: SEM images of (a) 2 % and (b) 4 % Li-doped ZnO compounds. (c) Estimation of the mean grain size of 2 % Li-doped ZnO compound by fitting the grain size distribution to the Gaussian curve. (d) EDS spectrum for the 4 % Li-doped ZnO compound. The positions for

Figure 5 :

 5 Figure 5: (a) Temperature variation of magnetization measured under zero-field cooled condition and an applied field of 500 Oe (b) Magnetization versus field curve measured at 3 and 300 K for ZnO compound.

Figure 6 :

 6 Figure 6: Temperature variation of magnetization of 2, 4, and 6% Li-doped ZnO samples measured under an applied field of 500 Oe. The filled symbol represents zero-field cooled M-T and the open symbol represents field-cooled M-T data.

Figure 7 :Figure 8 :

 78 Figure 7: Field variation of magnetization for Zn1-xLixO (x=0.02, 0.04, and 0.06) compounds measured at (a) 300 K and (b) 3 K. The inset in figure (b) indicates the coercivity of Li-doped ZnO compounds at 3K.

  experimental results on Li-doped ZnO compounds may fall in line with the predictions of this theory. Our experimental observations show that Li incorporation in ZnO lattice leads to d 0 ferromagnetic ordering at room temperature. With the increase of Li doping, a greater number of holes are created and thus it may also create Ruderman-Kittel-Kasuya-Yoshida type oscillations in the couplings, leading to the appearance of spin glass phase, as observed from magnetic irreversibility behavior between ZFC and FC M-T curves. Moreover, RKKY type oscillations in the couplings gradually reduces the magnetic ordering, resulting a decrease of the saturation magnetization with the increase of Li concentration, as observed from the M-H measurement.

  ZnO compounds, i.e. Zn1-xLixO (x=0, 0.02, 0.04 and 0.06) samples, prepared by solid-state reaction route method. From the study of crystal structure by X-ray diffraction (XRD) patterns, it is evident that the prepared materials have been formed in single-phase of the hexagonal wurtzite structure. The refinement of the XRD patterns suggests that there are only small changes in the lattice parameters upon Li-incorporation in ZnO. The average crystallite size (SC) was found to be in the range of 35-50 nm. The microstructural study by scanning electron microscope reveals the uniform morphology of the grains in the range of 50-70 nm. The energy dispersive spectrum indicates that no unwanted ferromagnetic impurities have crept into the final prepared samples. The measurement of the temperature (T) variation of magnetization (M) with SQUID magnetometer indicates that un-doped ZnO exhibits diamagnetic property but all Li-doped compounds exhibit room-temperature ferromagnetism and with a magnetic irreversibility behavior in zero-field cooled and field cooled M-T data. The measurements of hysteresis curves (M-H) at 3 and 300 K indicate that Li-doped samples exhibit ferromagnetic loops with ultra-soft coercivity (~50 Oe) and the maximum saturation magnetization (0.10 emu/gm) was observed for x= 0.02 sample. The value of MS decreases with an increase in Li concentration. Thus, from the results of M-T and M-H measurements, we can conclusively say that Li-doped compounds exhibit ferromagnetic behavior with a transition temperature much beyond room temperature and co-existence of low-temperature spin-glass phase.