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The ﬁnite temperature trapped dipolar Bose gasR.N.Bisset,D.Baillie,and P.B.BlakieJack Dodd Centre for Quantum Technology,Department of Physics,University of Otago,Dunedin,New Zealand.We develop a ﬁnite temperature Hartree theory for the trapped dipolar Bose gas.We use this theory to study thermal effects on the mechanical stability of the system and density oscillating condensate states.We present results for the stability phase diagram as a function of temperature and aspect ratio.In oblate traps above the critical temperature for condensation we ﬁnd that the Hartree theory predicts signiﬁcant stability enhancement over the semiclassical result.Below the critical temperature we ﬁnd that thermal effects are well described by accounting for the thermal depletion of the condensate.Our results also show that density oscillating condensate states occur over a range of interaction strengths that broadens with increasing temperature.PACS numbers:03.75.Hh,64.60.MyI.INTRODUCTIONA signiﬁcant new area of interest in ultra-cold atomic gases is the study of systems in which the particles interact via a dipole-dipole interaction (DDI)[1].This interest is be-ing driven by a broad range of proposed applications from condensed matter physics to quantum information,e.g.see [2].Experimental progress in the quantum degenerate regime has been driven by seminal work with 52Cr [3],which was Bose condensed in 2005,and more recently the realization of Bose-Einstein condensates of 164Dy [4]and 168Er [5].Po-lar molecules,which have DDIs several orders of magnitude larger than those of the atomic gases,have already been pro-duced in their ground rovibrational state [6,7],and steady progress is being made towards cooling these into the degen-erate regime.We also note the recent achievement of a degen-erate Fermi gas of 161Dy [8].The DDI is long-ranged and anisotropic with both attractive and repulsive components.Therefore,an important consider-ation is under what conditions the system is mechanically sta-ble from collapse to a high density state.Theoretical studies on zero temperature dipolar condensates reveal a rich stabil-ity diagram where,due to the DDI anisotropy,the stability is strongly dependent on the geometry of the trapping potential and the properties of the short ranged (contact)interactions [9–12].Another interesting theoretical observation is that for appropriate parameters (near instability)the condensate mode exhibits spatial oscillations and has a density maximum away from the minimum of the trapping potential [11–14].How-ever,evidence for this density oscillating state has yet to be observed in experiment.In this work we study the properties of a trapped dipolar Bose gas at ﬁnite temperature –a regime largely unexplored in theory and experiments.In previous work [15]we stud-ied the stability of a normal Bose gas (i.e.above T c )using a self-consistent semiclassical approximation.In this work we extend this study to below T c and to include quantum pressure (i.e.beyond-semiclassical effects)by numerically solving for the condensate and its ing this theory we study the crossover from the high temperature (above T c )to zero temperature (pure condensate)stability.Our results reveal that beyond semiclassical effects play a signiﬁcant role above T c in oblate geometry traps and enhance the stability region,andthat the double instability phase diagram in this trap geome-try (predicted in [15])remains prominent.We also study the behavior of the emergent biconcave condensate (density oscil-lating ground state)in the ﬁnite temperature regime,and ﬁnd that thermal effects enhance the density oscillation and en-large the parameter regime over which this type of state exists.We demonstrate that the below T c temperature dependence of the stability boundary is well-characterized by a simple model that accounts for the thermal depletion of the condensate.II.FORMALISM AND NUMERICAL IMPLEMENTATIONA.FormalismHere we consider a set of particles of mass M conﬁned in a cylindrically symmetric harmonic potentialU tr (x )=12M [ω2ρ(x 2+y 2)+ω2z z 2],(1)of aspect ratio λ=ωz /ωρ,with z and ρrepresenting the axial and radial directions,respectively.We take the particles to have dipole moments polarized along the z axis by an external ﬁeld,such that the DDI potential between particles isU dd (r )=C dd 4π1−3cos 2θ|r |,(2)where C dd =µ0µ2(=d 2/ 0)is the magnetic (electric)dipole-dipole interaction strength and θis the angle between the z di-rection and the relative separation of the dipoles (r =x −x ).It is easy to extend our calculations to include local (con-tact)interactions,however here we focus on the case of pure dipole-dipole interactions,as has been realized in experiments by use of a Feshbach resonance (e.g.see [16]).The Hartree formalism we employ (see Appendix A for a discussion of the relation to Hartree-Fock theory and relevant terms neglected)involves solving for the system modes using the non-local equationj u j (x )= − 22M∇2+V eﬀ(x )u j (x ),(3)a r X i v :1207.1929v 1 [c o n d -m a t .q u a n t -g a s ] 9 J u l 20122whereV eﬀ(x )=U tr (x )+d x U dd (x −x )n (x ),(4)n (x )=jN j |u j (x )|2,(5)are the effective potential and total density,respectively,with N j =[e β( j −µ)−1]−1the equilibrium (Bose-Einstein)oc-cupation of the mode,β=1/k B T the inverse temperature,and µthe chemical potential.Equations (3)-(5)are solved self-consistently while the chemical potential is adjusted to ensure that the desired total number N = d 3x n (x )is ob-tained.Below the critical temperature T c a condensate forms in the lowest mode u 0(x )with N 0∼N ,but the theory,as written in Eqs.(3)-(5),requires no additional adjustment to account for the condensate (due to our neglect of exchange)and smoothly transitions across T c .We emphasize that our motivation for using this theory is that it includes the domi-nant direct interactions and the full discrete character of the low energy modes,yet is more computationally efﬁcient than Bogoliubov-based approaches.This enables us to study chal-lenging problems that have not been explored,in particular ﬁ-nite temperature mechanical stability,in which obtaining con-vergent self-consistent solutions is demanding and time con-suming.Our numerical approach builds on various develop-ments (particularly those described in [17])and includes a number of features to aid calculations in the ﬁnite tempera-ture regime where interaction effects dominate (see Appendix B for details).The neglect of dipole exchange is consistent with other work on ﬁnite temperature bosons [18]and zero temperature studies of fermion stability [19].We would like to note that there is some justiﬁcation for this approximation.Studies on a normal trapped dipolar Fermi gas suggest that exchange inter-actions will quantitatively,but not qualitatively,affect stabil-ity [19,20].Indeed,the thermodynamic study of that system presented in [21]found that exchange interactions are typi-cally less important than direct interactions except for traps that are close to being isotropic.Similarly,Ticknor studied the quasi-two-dimensional Bose gas using the Hartree-Fock-Bogoliubov-Popov (HFBP)meanﬁeld theory [22]and found that exchange terms were generally less important than direct terms.III.RESULTSA.Comparison to previous calculationsTo benchmark our Hartree calculations we perform a quan-titative comparison to the HFBP calculations that Ronen et al.[18]performed for the three-dimensional trapped Bose gas at ﬁnite temperature.In Secs.III A 1and III A 2we make this comparison for two different sets of results from [18].We note that those HFBP calculations excluded thermal exchange interactions,although they did include condensate exchange interactions (exchange interaction of condensateatoms on the thermal excitations)[23].We extended ourHartree algorithm to include condensate exchange but found it made negligible difference to the predictions and do not in-clude results with this term here.1.Condensate FractionThe results of the ﬁrst comparison we perform are presented in Fig.1(a).There we compare the condensate fraction,as a function of temperature,for a system with λ=7.We ob-serve that the Hartree and HFBP theories predict an appre-ciably lower condensate fraction than the ideal case,and are in very good agreement with each other over the full tem-perature range considered.The low energy excitations of a Bose-Einstein condensate are quasi-particles,which are ac-curately described by Bogoliubov theory (such as the HFBP theory),however the thermodynamic properties of the system are dominated by the single particle modes (e.g.see [24]).A comparison of the Bogoliubov and Hartree-Fock spectra of a T =0dipolar Bose-Einstein condensate (BEC)was made in [17].That comparison revealed that the spectra were al-most identical,except for low energy modes with low values of angular momentum,where small differences in the modeFIG.1.(a)Condensate fraction and (b)density oscillation contrast (see text)for a dipolar BEC in a λ=7pancake trap.Hartree re-sults (pluses),HFBP results (solid lines),ideal gas result (dashed line).HFBP data corresponds to results shown in Figs.5and 6of Ref.[18].Other parameters:{ωρ,ωz }=2π×{100,700}s −1,N =16.3×10352Cr atoms with contact interactions tuned tozero.T 0c =3 N/ζ(3) ω/k B is the ideal condensation tempera-ture,where ω=3 ω2ρωz and ζ(α)is the Riemann zeta function with ζ(3)≈1.202.2.Density Oscillating Ground StatesAn interesting feature of dipolar condensates is the occur-rence of ground states with density oscillation features,where the condensate density has a local minimum at trap center.For3 a cylindrically symmetric trap these states are biconcave(redblood cell shaped–surfaces of constant density are shown inFig.6)ﬁrst predicted for T=0condensates in Ref.[11].In the purely dipolar case such biconcave states occur undercertain conditions of trap and dipole parameters,but notablyonly forλ 6and for dipole strengths close to instability.In[18]the HFBP technique was used to assess the effect oftemperature on the density oscillating states.This was char-acterized by the contrast,a measure of the magnitude of thedensity oscillation,deﬁned asc=1−n(0)n max,(6)where n(0)is the density at trap center and n max is the maxi-mum density of the system.In Fig.1(b)we compare our Hartree and HFBP theories for the contrast.This comparison reveals some small residual dif-ferences between the theories,however the results are in rea-sonable agreement and both predict that the contrast goes to zero(i.e.the condensate returns to having maximum density at trap center)at T≈0.65T0c.B.Mechanical stabilityOurﬁrst application of the Hartree theory is to study the ﬁnite temperature mechanical stability of a trapped dipolar Bose gas.To do this we construct a phase diagram for the range of dipole strengths for which the gas is stable for a num-ber of different trap geometries.Such stability properties,and the dependence on interactions and trap geometry,have been measured accurately in the dipolar system in the zero temper-ature limit(e.g.see[16]).We note theoretical studies[25–27] showing the important role of temperature on the observed stability of7Li condensates[28],which have an attractive con-tact interaction.1.Locating the stability boundaryWe consider a trapped sample ofﬁxed mean number N and wish to determine the values of the dipole interaction param-eter for which the system is mechanically stable as a function of temperature.In doing so we construct a phase diagram in{C dd,T}-space that indicates the stable region.In prac-tice we locate the stability boundary(i.e.a curve)that sep-arates the stable and unstable regions.Our procedure to ob-tain this boundary involves a computationally intensive search through parameter space toﬁnd the self-consistent solutions on the verge of instability.Determining the stability boundary forﬁxed mean number N complicates this process:since we work in the grand-canonical ensemble where the proper vari-ables are{µ,C dd,T},an additional iterative search over the parameterµis required toﬁx N to the desired target number. In Fig.2we provide some examples to illustrate how we identify the value of the DDI at the stability boundary for a gas with(target number)N=2×105atoms at a particularFIG.2.Locating instability(upper subplots):(a)The total num-ber of atoms of the self-consistent Hartree solution versus chem-ical potential forλ=1/8,k B T=40 ωand C dd=7×10−4 ωa3ho(dashed,case A),1.5×10−4 ωa3ho(solid,case B), with a ho=Each line terminates at the point of in-stability and occurs at the respective critical number N crit.(b) Same results as in(a)but plotted against 0−µ.The dotted line represents the target number,in this case N=2×105.Den-sity Proﬁles(lower subplots):Solid(dashed)line represents the ra-dial(axial)density n,higher curves are near the stability bound-ary.λ=1,N=2×105.(c)T/T0c=0.82(N0/N≈0.43) and C dd/4πC0={3.65×10−4(gray),1.83×10−4(black)}.(d)T/T0c=1.27and C dd/4πC0={2.91(gray),1.22(black)}. We have introduced the interaction strength unit C0= ωa3ho/6√N which is convenient for cases where N isﬁxed,and allows our sub-sequent results to be directly compared to those in[15].temperature.To do this we show the dependence of total atom number onµfor two different values of the DDI[Fig.2(a)]. For both curves the total number increases as we move along these curves until some maximum value N crit is reached at which the system becomes unstable.The non-monotonic be-havior of these curves arises because the ground state energy 0changes as the number of atoms increases,and hence the role of DDIs increases.For this reason we also show the same two cases,but as a function of 0−µ,in Fig.2(b).The sharp cusps in Figs.2(a)and(b)correspond to the point where the system condenses[i.e.where 0−µ≈0].The dependence of 0on N0is strongly dependent on the trap ge-ometry,and for the cases we consider here withλ=1/8, 0 decreases with increasing N0.This is because the head-to-tail character,in the cigar geometry,emphasizes the attractive part of the DDI so that as the condensate number increases, 0 (≈µ)decreases.a stringent numerical task and requires careful convergence tests.For condensates with contact interactions this type of numerical instability analysis was applied in Refs.[25–27](also see Ref.[15]).In Fig.2(a)the instability point occurs at the end of the upper horizontal plateau in the N versus µcurves (compare to Fig.1of [25]).We show examples of the spatial density proﬁles for a spherical trap in Figs.2(c)and (d).The system considered in Fig.2(c)is condensed,while that considered in Fig.2(d)is above the critical temperature.For both cases a result is shown that is well inside the stable region (black curves)and near the stability boundary (gray curves).Despite a large difference in the density scales of the two regimes they both exhibit a similar sharpening of the density proﬁle near instability.An additional consideration emerges for stability calcula-tions below T c in regimes where the condensate is in a density oscillating state.Here the ﬁrst mode to go soft (and then de-velop imaginary parts)as the stability boundary is reached is a m =0quasi-particle mode [29],where m is the angular mo-mentum projection quantum number (so called angular roton mode [11]).This instability is not revealed in the Hartree exci-tations,and as we solve for the condensate in the m =0space (see Appendix B),the condensate does not exhibit numerical instability.Thus in cases where the condensate exhibits a den-sity oscillating state we perform a Bogoliubov analysis of the condensate mode (within the effective potential of the self-consistent Hartree solution)to determine if any m =0modes have become unstable [30].2.Stability above T cIn Fig.3we show our results for the stability of the normal phase.In previous work we examined this regime using a semiclassical Hartree approach in which the density isn (x )=λ−3dB ζ3/2 e β[µ−V eff (x )],(7)where V eﬀ(x )is the effective potential calculated using n (x )[see Eq.(4)],ζα(z )= ∞j =1z j /j αis the Bose function,andλdB =h/√2πMk B T .The semiclassical results are shown as solid lines in Fig.3.FIG.3.(Color online)Stability regions in DDI-temperature space.Shaded regions indicate stability for each geometry,from top to bot-tom λ={8,4,2,1,1/2,1/8},the geometric mean trap frequency is ﬁxed and N =2×105.Actual data points represented by symbols while the shading of the stable regions interpolates to guide the eye,the semiclassical model is given by the solid curves.Error bars rep-resent the 1σspread in the convergence test (see Appendix B 3for more details).We observe that as a general trend the stability region grows with increasing λ.The strong geometry dependence of these results arises from the anisotropy of the dipole interaction:In oblate geometries (λ>1)the dipoles are predominantly side-by-side and interact repulsively (stabilizing),whereas in pro-late geometries (λ<1)the attractive (destabilizing)head-to-tail interaction of the dipoles dominates (a similar geometry dependence is observed for the stability of T =0dipolar con-densates [11,12]).A primary concern is the nature of beyond semiclassi-cal effects,i.e.what differences emerge from our diagonal-ized Hartree theory over the semiclassical formulation.Most prominently in the results of Fig.3we observe that while the Hartree and semiclassical stability boundaries are in good agreement for prolate geometries,in oblate traps the Hartree results are signiﬁcantly more stable.This difference between the boundaries predicted by the two theories increases with increasing λ.This observation is surprising because our cal-culation is for a rather large number of atoms (N =2×105),where the semiclassical approximation would normally be ex-pected to furnish an accurate description of the above T c be-havior.We attribute this failure of the semiclassical theory to its inappropriate treatment of the interactions between the low energy modes [31].The nature of the DDI,when tightly con-ﬁned along the polarization direction,has been extensively studied in application to pure BECs [14,32],where it has been shown that it confers additional stability on the system,as veriﬁed in recent experiments [33].This arises from a con-ﬁnement induced momentum dependence of the interaction:the interaction is repulsive (stabilizing)for low momentum interactions,but decays to being attractive with a character-istic wavevector k ∼1/a z set by the z conﬁnement length5a z=zNotably these features of the conﬁned inter-action mediate BEC instability through the softening of radi-ally excited modes with a wavelength∼a z[32,34–37].It is not clear that these conﬁnement effects will be applica-ble at a modestly oblate trap withλ=8,however numerical studies have revealed that quasi-particle modes with a wave-length∼a z soften in a BEC withλ=7[11].Within the limited range of results we have forλ>1we see evidence consistent with conﬁnement induced effects playing an impor-tant role in the above T c Hartree calculations.Notably,that the relative difference between the stability boundaries of the Hartree and semiclassical calculations scale with1/a2z.Also, when the system is unstable,during the self-consistency it-erations(prior to collapse)strong radial densityﬂuctuations develop in the systemA key prediction from our semiclassical study[15]is a dou-ble instability feature in oblate trapping geometries arising from the interplay of thermal gas saturation and the anisotropy of the DDI.Our Hartree calculations in this oblate regime, despite shifting the stability boundary from the semiclassical prediction by a considerable amount,reveal that the double instability feature is robust to beyond-semiclassical effects.A prominent feature of the semiclassical calculation is that the stability curves for the purely dipolar gas terminate at the critical point with C dd=0(i.e.predicting that without con-tact interactions only an ideal gas is stable below T c).This oc-curs because the local compressibility at trap center diverges at the critical point and the gas is unstable to any attractive in-teraction(see[15]).In the beyond-semiclassical calculations the trap provides aﬁnite momentum cutoff that prevents the divergence of compressibility,and thus the system has aﬁ-nite residual stability at and below T c(which we consider in Sec.III B3).3.Stability below T cIn Fig.4we consider the stability below T c where the semi-classical model does not apply.These results are identical to those shown in Fig.3,but the below T c details are revealed us-ing a logarithmic vertical pared to the above T c gas the condensate is rather fragile,with the critical DDI strength deﬁning the stability boundary decreasing by∼3to4orders of magnitude.In the zero temperature limit our results agree with previ-ous calculations based on solving the Gross-Pitaevskii equa-tion[11].This agreement is expected as the two theories are identical when the excited modes have vanishing population. For a pure condensate,the critical DDI strength depends on the condensate number and trap geometry according to[11]C dd=F(λ)N0,(T=0)(8)with F(λ)a rather interesting function of trap geometry alone, as characterized in Fig.1of[11][41].More generally,beyond the case of pure DDIs,F also depends on the contact interac-tion strength,e.g.see[12,37].FIG.5.(Color online)Stability boundary scaling.The stability boundary results(symbols)have been taken from Fig.4forλ= {8,1,1/8}(top to bottom).Dashed line prediction is based on a non-interacting N0scaling(see text)and the solid line uses the N0calcu-lated from the Hartree solutions.As temperature increases,but focusing on T<T0c,we ob-serve in Fig.4that the stability boundary increases signiﬁ-cantly.This occurs because as the temperature increases the condensate is thermally depleted.Indeed,by simply account-ing for the thermal depletion we can immediately extend result(8)to predict the critical value of the DDI atﬁnite temperatureC dd(T):C dd(T)=F(λ)N0(T)=C dd(0)NN0(T),(9)where the last expression is obtained using N0(T=0)=N. Equation(9)predicts that the stability atﬁnite temperaturewas in Ref.[18][which we reproduce in Fig.1(b)].That study considered a single line (at ﬁxed C dd and N and varying T )through the phase diagram,and showed that biconcavity per-sisted at small ﬁnite temperatures (T 0.25T 0c ),but then was rapidly washed out as temperature increased further.Using our Hartree theory we provide a broad characteriza-tion of the thermal effects on biconcavity.We focus on the case λ=8,which supports a biconcave condensate at T =0.In Fig.6(a)we present contours of biconcave contrast [as de-ﬁned in Eq.(6)]over the entire range of parameters where this state is stable.These results show that biconcavity is not destroyed as temperature increases.Instead the parameter re-gion over which biconcavity occurs grows,with large bicon-cave contrasts emerging at higher temperature.The general trends seen can be understood by considering the thermal de-pletion of the condensate,using similar arguments to those made to obtain Eq.(9):as the temperature increases the value of C dd required for the condensate to exhibit a biconcave den-sity proﬁle should increase in a manner that is approximately inversely proportional to the condensate occupation.Thus,the washing out observed in [18][our Fig.1(b)]arises because they considered C dd ﬁxed.Thermal depletion of the conden-sate is not sufﬁcient to explain all aspects observed in our re-sults,e.g.the deepening of the biconcave contrast that devel-ops at higher temperatures in Fig.6(a).This arises from ad-ditional effects of the thermal interaction with the condensate,e.g.small changes in the aspect ratio of the effective poten-tial that the condensate experiences can signiﬁcantly changeeral trends seen can be understood by considering the ther-mal depletion of the condensate,using similar arguments to those made to obtain Eq.(9):as the temperature increases a the value of C dd required for the condensate to exhibit a bi-concave density proﬁle should increase in a manner that is approximately inversely proportional to the condensate occu-pation.Thus,the washing out observed in [18][our Fig.1(b)]arises because they considered C dd ﬁxed.Thermal depletion of the condensate is not sufﬁcient to explain all aspects ob-served,e.g.the deepening of the biconcave contrast.This arises from additional effects of the thermal interaction with the condensate,e.g.small changes in the aspect ratio of the ef-fective potential from the condensate can signiﬁcantly change the contrast (c.f.Fig.1of Ref.[11]).In Fig.6(lower)we show two examples of the biconcave density proﬁles at different temperatures.Case B displays the very pronounced biconcavity for a system at T ≈0.9T 0c ,where the condensate fraction is N 0/N ≈0.07.IV .CONCLUSIONSIn this paper we have developed a Hartree theory for a trapped dipolar Bose gas that can be applied to make predic-tions above and below the critical temperature T c for conden-sation.We have used this theory to quantify the role of ther-mal ﬂuctuations on the mechanical stability of the cloud,and present results for the stability phase diagram as a function of 6810ρ/a h o(b)FIG.6.(Color online)Biconcave characteristics for λ=8and N =2×105at ﬁnite temperature.(a)Stability diagram for λ=8from Fig.3with biconcave contrast contours {0,0.05,0.1,0.15,0.2,0.25}(bottom to top)added.The solid curves are interpolations be-tween the calculated contours points.The white dotted line marks where we terminate the contours due to the condensate fraction be-coming negligibly small.Triangles indicate the stability boundary from Fig.5.Inset:Magniﬁcation of the high temperature region.(b)Radial densities for phase space points marked by A and B of the upper ﬁgure.A:T/T 0c=0.0910,C dd /4πC 0=0.00268and N 0/N =1.00(thermal depletion <1%).B:T/T 0c =0.910,C dd/4πC 0=0.0291and N 0/N =0.0716.Solid (dashed)lines represent the total (condensate)density.Insets:corresponding sur-face contour at 67%of the peak density.temperature and aspect ratio.Our results show that the ther-mal depletion of the condensate can lead to an enhancement of the parameter regime over which biconcave density oscil-lations are found.Furthermore,a large thermal cloud may actually enhance the biconcave contrast making direct imag-ing of an in situ blood cell more feasible,see Fig.6(lower).Above T c we ﬁnd that the results of our theory predict signif-icant corrections to the stability boundary over the equivalent Hartree semiclassical theory.Most notably,the semiclassical theory underestimates the size of the stability region for oblate FIG.6.(Color online)Biconcave characteristics for λ=8and N =2×105at ﬁnite temperature.(a)Stability diagram with bi-concave contrast contours {0,0.05,0.1,0.15,0.2,0.25}(bottom to top)added.The solid curves are interpolations between the cal-culated contour points.The white dotted line marks where we ter-minate the contours due to the condensate fraction becoming negli-gibly small.Triangles indicate the stability boundary from Fig.4.Inset:Magniﬁcation of the high temperature region.(b)Radial densities for phase space points marked by A and B in (a).A:T/T 0c =0.0910,C dd /4πC 0=0.00268and N 0/N =1.00(ther-mal depletion <1%).B:T/T 0c =0.910,C dd /4πC 0=0.0291and N 0/N =0.0716.Solid (dashed)lines represent the total (conden-sate)density.Insets:corresponding surface contours at 67%of the peak density.the contrast (c.f.the strong dependence of biconcavity on trap aspect ratio near λ=8in Fig.1of Ref.[11]).In Fig.6(b)we show two examples of the biconcave density proﬁles at different temperatures.Case B displays the verypronounced biconcavity for a system at T ≈0.9T 0c ,wherethe condensate fraction is N 0/N ≈0.07.IV .CONCLUSIONSIn this paper we have developed a Hartree theory for a trapped dipolar Bose gas that can be applied to make predic-。

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Dielectric and magnetic properties of Fe-and Nb-doped CaCu3Ti4O12 R.K.Grubbs,E.L.Venturini,P.G.Clem,J.J.Richardson,B.A.Tuttle,and G.A.Samara Sandia National Laboratories,Albuquerque,New Mexico87185,USA͑Received20January2005;revised manuscript received20June2005;published23September2005͒Detailed studies of the properties of ceramic CaCu3Ti4O12͑CCTO͒have clariﬁed the physics of this inter-esting material and revealed several features not reported before.The dielectric relaxational properties of CCTO are explained in terms of a capacitive-layer model,as for an inhomogeneous semiconductor,consisting of semiconducting grains and insulating grain boundaries as also concluded by others.The kinetics of the main ͓low-temperature͑T͔͒relaxation reveal that two different thermally activated processes in CCTO grains control the dynamics.A likely candidate defect responsible for the two processes is the oxygen vacancy which is a double donor.A higher-T relaxation is determined by grain boundary conduction.Both Nb and Fe doping lowered both the apparent dielectric constant␧Јand the dielectric loss,but increased Fe doping led to more dramatic effects.At3at.%Fe doping,the anomalous␧Ј͑T͒response was removed,making the CCTO an intrinsic,very-low-loss dielectric.The intrinsic␧Ј͑Ӎ75͒and its T dependence are measured and shown to be largely determined by a low-lying soft TO phonon.At low T,cubic CCTO transforms into an antiferromagnetic phase at T N=25K.T N is essentially independent of Nb doping͑up to4at.%͒and of hydrostatic pressure͑up toϳ7kbar͒,but decreases signiﬁcantly with Fe doping.Analysis of the high-T dependence of the magnetic susceptibility provided insight into the role of Fe as a dopant.Finally,an␧Ј͑T͒anomaly associated with the onset of antiferromagnetic order has been discovered,providing evidence for coupling between the polarization and sublattice magnetization.The possible origin of this coupling is discussed.DOI:10.1103/PhysRevB.72.104111PACS number͑s͒:77.22.Ch,75.30.Kz,81.40.RsI.INTRODUCTIONThe“unusual”dielectric properties of CaCu3Ti4O12͑CCTO͒have been of much recent interest.The material has a distorted,complex cubic perovskite͑AB O3͒-like structure with a large unit cell͑aϷ7.4Å͒.1,2The Cu ions form a square planar lattice on the A sites and are bonded to four oxygen atoms each,causing tilting of the TiO6octahedra.Subramanian et al.1were theﬁrst to report a very large ͑Ͼ10000͒and nearly temperature-͑T−͒independent appar-ent static dielectric constant␧Ј,for ceramic samples of CCTO at Tജ300K.Subsequent work on ceramic2–5as well as single-crystal2samples conﬁrmed this early work and re-vealed additional features.In particular,belowϳ200K␧Јexhibits relaxational behavior,ultimately dropping to a value ofϳ100at the lowest T’s with the crystal structure remain-ing cubic and centrosymmetric down to at least35K.While dipole relaxation models associated with the com-plex crystal structure of CCTO have been suggested to ex-plain the dielectric response,2,3it is now generally agreed that the observed behavior is not intrinsic.Rather,the dielec-tric response is due to barrier-layer capacitances associated with one or more of the following:grain boundaries,twin boundaries,dislocation networks,and Schottky barrier or in-terfacial polarization effects.1,4,6–10A detailedﬁrst-principles study6of the structure and lattice dielectric response of CCTO did not reveal any unusual features in the zone-center optical phonons and estimated the intrinsic lattice contribu-tion to␧Јto beϳ40.Cohen et al.9explored various sce-narios of conducting and insulating structures in an inhomo-geneous solid that can lead to very large␧’s and that could agree with the observed response of CCTO.They considered cases where the bulk of the sample͑or grain͒is either con-ducting or nearly so and the grain and domain boundaries are insulating,or vice versa.These studies show that large␧’s result either when the conducting regions approach a perco-lation threshold or when they percolate but are blocked at the sample surface or electrode interface.In CCTO oxygen va-cancies and/or impurities can lead to conductivity in either the bulk͑grains͒or grain boundaries.Adams et al.7sug-gested an explanation for the behavior of CCTO in terms of conducting grains and insulating grain boundaries.Strong support for this suggestion was provided by Chung et al.10in the most deﬁnitive work thus far on undoped ceramic CCTO. Through a combination of microcontact I-V measurements, Kelvin probe force microscopy,and resistivity and thermo-electric power measurements on individual grains,as well as across grain boundaries,these authors have determined that ͑1͒the grains are conducting and are n type,and͑2͒a large potential barrier exists at the grain boundaries reﬂecting their insulating character.It has long been known11that CCTO exhibits an antifer-romagnetic͑AFM͒transition atϳ25K.The transition re-gion has been recently studied by neutron diffraction12and electron paramagnetic resonance͑EPR͒,13and the antiferro-magnetism has been explained in terms of a double primitive cell in which each Cu-Cu nearest-neighbor pair has antipar-allel spins.The behavior of the dielectric response near this transition apparently has not yet been studied.The purpose of the present work is to investigate the ef-fects of doping͑Nb and Fe͒and hydrostatic pressure on the dielectric properties and AFM transition of CCTO in order to shed additional light on the interesting properties of this complex oxide.Also,we sought evidence͑which we found͒for a possible coupling between the magnetization and polar-ization below the AFM transition or Neel temperature T N. Consequently,a considerable part of our effort was devotedPHYSICAL REVIEW B72,104111͑2005͒to the low-temperature properties where the response of CCTO can be expected to become essentially intrinsic.Dop-ing with iron ͑Fe ͒allowed us to reach the intrinsic regime where we found the behavior of ␧Ј͑T ͒to be different from that suggested by earlier work.Iron doping is also found to produce remarkably large changes in the overall dielectric response.In what follows we shall ﬁrst give a brief descrip-tion of the experimental details in Sec.II followed by a pre-sentation and discussion of the results in Sec.III.II.EXPERIMENTAL DETAILSA.SamplesCeramic CaCu 3Ti 4O 12͑pure,as well as Fe or Nb doped ͒pellets were prepared using a solid-state reaction and sinter-ing process.Precursor powders of CaCO 3͑ജ99.95%Ald-rich ͒,TiO 2͑ജ99.8%Aldrich ͒,and CuO ͑ജ99.99%Aldrich ͒were directly combined in the necessary stoichiometric ratios while doped samples employed Fe 2O 3͑ജ99.3%Baker Chemical ͒as a powder or Nb n -butoxide ͑ജ99%Inorgtech ͒in ethanol,as dopant precursors.Thorough mixing was achieved by milling the powders in ethanol or acetone with zirconia media for no less than 12h in a rolling mill.The dried mixed powders were calcined at 1173K ͑900°C ͒for 1h before being milled a second time.The dried powders wereuniaxially cold pressed into 12in.pellets under a load of 2000lb ͑ϳ10ksi pressure ͒.The resulting pellets were then sin-tered in air on top of Pt setters ͑to avoid reaction with the ceramic alumina furnace boats ͒using the schedule 10K/min to 1273K ͑1000°C ͒,6h hold,10K/min to 1373K ͑1100°C ͒,6hour hold,and then furnace cooled to room temperature.After sintering,the pellets were conﬁrmed to be polycrystalline,predominantly single-phase,CaCu 3Ti 4O 12by ␪-2␪x-ray diffraction ͑XRD ͒.Figure 1shows a typical microstructure of CCTO revealing a broad distribution of grain sizes.The picture in Fig.1shows an enhanced contrast scanning electron microscope ͑SEM ͒image with the light colored areas identiﬁed as Cu rich.Interestingly,no second phases were observed on logarithmic-scale XRD scans of CCTO.To control the defect chemistry of perovskite AB O 3ox-ides such as BaTiO 3,SrTiO 3and Pb ͑Zr,Ti ͒O 3͑PZT ͒,a va-riety of dopants have been developed to occupy the A -and B -site cation positions.14–16Tailoring materials by way ofdefect chemistry,including cation and oxygen vacancy con-centration,with aliovalent cation dopants enables control of their semiconducting behavior,including free carrier type,carrier concentration,and hence,material resistivity and real and imaginary permittivity.In this work,CaCu 3Ti 4O 12was modeled as a 4͑AB O 3͒perovskite where ideal cation charges would be +2for the A -site and +4for the B -site cations.Similar to the presence of lead vacancies inducing oxygen vacancies and p -type behavior in PZT,copper volatility and reduction of Cu from +2to +1was anticipated to produce oxygen-vacancy-related conductivity in CCTO.16A number of dopants were selected on the basis of literature reports of n -and p -type perovskite dopants and prediction of dopant types by consideration of ionic radius and charge state.17Niobium is a commonly used n -type dopant,as Nb 5+͑ionic radius r =0.64Å͒it occupies a Ti 4+͑r =0.68Å͒site,while Fe 3+͑r =0.55Å͒was predicted to be a p -type dopant on the Ti 4+site.Samples were doped with 0–4%of Nb or Fe sub-stituting for Ti,as such doping levels are typically within the solid solution solubility limit for perovskites and were found to display single-phase CCTO x-ray diffraction patterns.For dielectric measurements,sections of the pellets were cut using a diamond saw ͑typically cut into 4ϫ4mm 2squares that were 0.50mm thick ͒,and polished.Consistent results came from samples cut from the center of the pellet.These will be denoted CCTO sample A .Similar,but lower-magnitude,dielectric responses were observed from samples cut from the end of the pellet.The dielectric response from an edge-cut sample is included in this study and is labeled CCTO sample B .Metal electrodes were rf magnetron sputter deposited on the samples.Initially,a thin ﬁlm of Cr,approxi-mately 650Åthick,was used as an adhesion metal before 1000Åof Au was deposited.The electroded samples were rigidly mounted between two copper paddles and a small quantity of silver paint was used to form an intimate contact between the Cu paddles and the metalized sample surface.The measurements were made with the sample placed in a He pressure cell that was in turn housed in a temperature Dewar.B.MeasurementsThe real ͑␧Ј͒and imaginary ͑␧Љ,or the loss factor tan ␦=␧Љ/␧Ј͒components of the dielectric function were mea-sured with an Agilent 4284A LCR meter as functions of temperature ͑5–400K ͒,hydrostatic pressure ͑0–7kbar ͒,and frequency ͑100Hz–1MHz ͒.No thermal hysteresis was ob-served for any of the samples used in this study.For the pressure experiments helium was the pressure-transmitting medium.A commercial superconducting quantum interfer-ence design magnetometer was used for the magnetic mea-surements at 1bar.The pressure dependence of T N was as-sessed from the pressure dependence of the anomaly of ␧Ј͑T ͒at T N .Analysis of ␧Ј͑T ͒at low temperatures ͑i.e.,in the intrinsic regime ͒required knowledge of the thermal expan-sion and volume compressibility of CCTO.The thermal ex-pansion below 300K was reported by Ramirez et al.3and the volume compressibility was assumed to be comparable to that of other complexperovskites.FIG.1.A typical microstructure of our ceramic CCTO samples as determined by SEM.GRUBBS et al.PHYSICAL REVIEW B 72,104111͑2005͒III.RESULTS AND DISCUSSIONA.Dielectric propertiesIn the course of this work we carried out detailed studies of the properties of undoped ceramic CCTO samples.The shapes and magnitudes of the ␧Ј͑T ͒and tan ␦͑T ͒curves as well as the frequency dispersions of the samples are very similar to the now familiar response of CCTO.2–5,8A typical ␧Ј͑T ͒response for CCTO is shown in Fig.2.However,our results revealed additional features in the relaxational dy-namics which are presented in Sec.III B.We also investigated the inﬂuence of hydrostatic pressure on the dielectric response.Figure 2shows that a pressure of 5kbar ͑very modest compression ͒has very little inﬂuence on ␧Ј͑T ͒,generally causing small decreases in the magnitude of ␧Јat ﬁxed T .The decrease in ␧Јdue to pressure is seen most clearly in the low-T regime depicted in the two insets.We note that the ␧Ј͑P ͒slopes at the two temperatures ͑20and 40K ͒are essentially independent of frequency.Over the narrow pressure ranges covered in the insets,one would expect the ␧Ј͑P ͒response to be linear.Thus,the nonlinear ␧Ј͑P ͒at the very lowest pressures at 20K may reﬂect the inﬂuence of the AFM phase ͑Sec.III D ͒.The weak pressure dependence of ␧Јobserved in CCTO is consistent with the barrier-capacitance model for the dielectric response as these modest pressures are not expected to signiﬁcantly change the prop-erties of the grains or grain boundaries and their thicknesses.As noted earlier,our doping studies with Fe and Nb were aimed at modifying the electronic transport in CCTO to con-trol the dielectric loss.Expecting that Fe 3+and Nb 5+would substitute primarily at the Ti 4+site,then Fe 3+would make the material p type and Nb 5+would make it n type.Figure 3shows the dielectric response of a 1at.%Nb doped sample.These results are qualitatively similar to those for the un-doped CCTO samples.A 4at.%Nb doped sample also showed similar results,but ␧Ј͑T ͒in the plateau region was lower than that for the 1%Nb sample.A decrease of ␧Јin the plateau regime with Nb doping is consistent with recent observations by Aygun et al.5that showed a decrease of ␧Јwith increased doping for samples doped with Nb and annealed in an oxygen atmosphere.We believe,as is also predicted from the grain and grain bound-ary capacitive-layer model,that the primary reason for this effect is an increase in grain boundary ͑oxide ͒layer thick-ness resulting in lower grain boundary capacitance,the change in grain resistivity with doping being relatively mod-est and inconsequential.The low-T dielectric properties of the 1and 4at.%Nb-doped samples are very similar to those of the undoped samples.The low-T results for the 1%Nb sample are shown in the inset in Fig.3͑a ͒.It is seen that the response at the lowest temperatures is still very dispersive,but additional features are revealed.The break in the ␧Ј͑T ͒slope at 24K,which also shows up as a slight bump in tan ␦͑T ͒,is a feature that is associated with the AFM transition at the Néel tem-perature,T N .We shall come back to this feature below.An-other feature in the low-T data is a relaxational process at ϳ10K.It is somewhat similar in shape to the anomaly at T N,FIG.2.Typical dielectric response of an undoped CCTO sample ͑sample B ͒at 1bar and under the inﬂuence of hydrostatic pressure.The insets show the pressure dependence of ␧Јat two low tempera-tures and the ﬁnite ⌬T suggests a nondipolarrelaxation.FIG.3.Temperature dependences of ␧Јand tan ␦for a 1at.%Nb-doped CCTO sample at different frequencies at 1bar.The inset in ͑a ͒shows details of the ␧Ј͑T ͒response to low temperatures and the ﬁnite ⌬T suggests a nondipolar relaxation.DIELECTRIC AND MAGNETIC PROPERTIES OF Fe-…PHYSICAL REVIEW B 72,104111͑2005͒but unlike the latter which is frequency independent,it ex-hibits strong frequency dispersion.It is likely due to a dipo-lar impurity or defect of unknown origin.The response of a 1at.%Fe-doped CCTO sample is shown in Fig.4.In Fig.4͑a ͒we note the very marked reduc-tion in the values of ␧Јin the plateau region ͑inset ͒—from many thousands for undoped and Nb-doped CCTO to several hundreds for the 1at.%Fe sample.Associated with the re-duced ␧Јis a large reduction in the peak value of tan ␦shown in Fig 4͑b ͒.This value goes from 2to 3for the samples in Figs.2and 3to ϳ0.7for 1at.%Fe.Despite these differences,the low-T relaxation labeled ͓A ͔in Fig.4͑b ͒occurs in the same temperature range of 50–250K as for the other CCTO samples.Fe doping causes additional features not observed in the other samples over the temperature range of the present mea-surements ͑but expected at higher T ’s,as will be discussed later ͒.Figure 4reveals a relaxational feature at higher T in both ␧Ј͑T ͒and in tan ␦͑T ͒,represented by the group of peaks labeled ͓B ͔.The dynamics of this feature will be discussed in Sec.III C below.Fe doping also strongly inﬂuences the low-temperature response.The low-T results for 1at.%Fe are shown in Fig.5.First,note the increase of ␧Јwith decreasing T above T N .Second,note that the dielectric loss shown in the inset is much lower and the frequency dispersion much less than for the other samples.Both properties suggest that this sample is closer to exhibiting intrinsic behavior at low temperatures ͑explained in Sec.III B below ͒.Also worth noting is that the relaxational process near 10K is still present in this Fe-doped sample,conﬁrming that it is associated with a dipolar entity characteristic of ͑ceramic ͒CCTO.Increasing the Fe dopant concentration from 1to 3at.%produces further remarkable changes in the dielectric re-sponse of CCTO.Figures 6–8show these results.Figures 6and 7show dielectric response of the 3at.%Fe-doped CCTO sample.There is a dramatic decrease in the magnitude of ␧Јcompared to the 1at.%Fe sample ͑cf.Fig.4͒.Both the values of ␧Јin the “plateau”region and ͑tan ␦͒max are an order of magnitude lower than those of the 1at.%Fe sample ͑and about two orders of magnitude lower than the undoped samples ͒.There is also a noticeable shift in the plateau re-gion to higher temperataures.The high-T relaxational peaks observed in Fig.6͑b ͒are similar to the ͓B ͔relaxations seen in the 1at.%Fe sample in Fig.4͑b ͒.The mid-temperature region of 50–250K for the 3at.%Fe CCTO sample is shown in Fig.7.It is seen that both the ␧Ј͑T ͒and tan ␦͑T ͒relaxations ͑labeled ͓A *͔for this sample͒FIG.4.Temperature dependences of ␧Јand tan ␦for a 1at.%Fe-doped CCTO sample at different frequencies at 1bar.The inset expands the ␧Ј͑T ͒response below 200K.FIG.5.A detailed view of the ␧Ј͑T ͒response of the 1at.%Fe-doped sample at low temperatures.The inset shows the tempera-ture dependence of the dielectricloss.FIG.6.Temperature dependencies of ␧Јand tan ␦for a 3at.%Fe-doped sample showing the suppression of the ␧Ј͑T ͒plateau be-low ϳ250K.The dielectric response of undoped CCTO at 10kHz is plotted for comparison.GRUBBS et al.PHYSICAL REVIEW B 72,104111͑2005͒have qualitatively different character from those observed in the 1at.%Fe sample in Fig.4͑b ͒͑relaxation labeled ͓A ͔͒as well as in the other CCTO samples studied.Note in particu-lar that the inﬂection point in the ␧Ј͑T ͒trace and the peak in the tan ␦͑T ͒occur at the same temperature as indicated by the double arrow on the 10kHz curve.This suggests that this particular relaxation is due to a dipolar entity in the sample and not due to extrinsic barrier-layer capacitances created between grains and grain boundaries,as was the case for all the other samples studied.As will be discussed in Sec.III B below,the ͓A *͔relaxation in Fig.7͑b ͒exhibits different ki-netics from those exhibited by the ͓A ͔relaxations in other samples.Figure 8shows the results for the 3at.%Fe CCTO sample below ϳ100K.Here,the ␧Јvalues are 25–30%lower than for the 1at.%Fe sample and the frequency dis-persion and dielectric loss are becoming vanishingly small.Thus,the dissipative mechanisms are essentially frozen and the dielectric response should be close to the intrinsic re-sponse of the sample.In this regard the increase in ␧Јwith decreasing T is characteristic of soft to mode behavior.The light dashed line below T N in Fig.8would be the dielectric response of the sample in the absence of AFM ordering.This extrapolation clearly shows that d ␧Ј/dT would go to 0as T →0K,as required by the third law of thermodynamics.We note that the ␧Ј͑T ͒anomaly associated with AFM transition is now seen at T N =21K.Clearly,there are signiﬁcant doping effects associated with the 3at.%Fe in CCTO that reﬂect important changes in the conductivity of either the grains and/or grain bound-aries,with increasing Fe concentration.We shall discuss these effects in Sec.III B.Figure 9provides a summary of the ␧Ј͑T ͒and tan ␦͑T ͒results for all of the samples used in this study at one fre-quency ͑10kHz ͒.Here,one can observe the remarkable in-ﬂuence that the addition of Fe has on the dielectric response of CCTO.All these results ͑except the 3at.%Fe-doped sample ͒can be qualitatively understood in terms of the con-ducting grain and insulating grain boundary model cited in the Introduction.The physics of this model has been known for a long time.In the case of the 3at.%Fe-doped sample,the relaxation around 150K in Fig.7is,as already noted,of dipolar origin,and it is seen only when ␧Ј͑T ͒and tan ␦͑T ͒approach intrinsic values.It may be due to an Fe-induced dipolar defect,or it may be present in all CCTO samples,but it is generally not observed in the dielectric data because it is hidden under the huge ␧Ј͑T ͒and tan ␦͑T ͒responses associ-ated with the barrier layers ͑compare,e.g.,Figs.3and 7͒.In the course of this work we noted a correlation between the magnitude of ␧Јin the plateau region and the peak value of tan ␦for the low-temperature ͓A ͔relaxation.It was appar-ent that a larger ␧Јin the plateau region correlated withaFIG.7.Dipolar relaxational response of the 3at.%Fe-dopedsample.FIG.8.The intrinsic dielectric response of the 3at.%Fe-doped sample showing soft-mode behavior and the ␧Ј͑T ͒anomaly at T N.FIG.9.Summary ﬁgure showing the inﬂuence of Nb and Fe doping on the dielectric response of CCTO.DIELECTRIC AND MAGNETIC PROPERTIES OF Fe-…PHYSICAL REVIEW B 72,104111͑2005͒larger value of the tan ␦peak maximum.This correlation is shown for our samples at a frequency of 10kHz in Fig.10.We have also included on this plot some of the data in the literature on ceramic samples that published both ␧Јand ͑tan ␦͒max at 10kHz.It is seen that all the data accurately follow the same trend which is qualitatively predicted by the barrier-layer-capacitance model.18The discussion thus far has dealt with ceramic samples.But what about single-crystal CCTO?Homes et al.2studied the dielectric response of single-crystal CCTO and found re-sults qualitatively similar to those for ceramic samples,but both the plateau values of ␧Јand the peak tan ␦values were much larger in magnitude.We have plotted their results at 10kHz as a solid triangle in Fig.10.Clearly,their datum point follows the trend observed for ceramic samples.The single crystal,of course,does not have the grains and grain bound-aries alluded to in the barrier-layer-capacitance model.How-ever,the single crystals do possess two interfaces and a bulk crystal interior which could display different resistivity char-acteristics and thus similar RC barrier properties to ceramic samples.It is also possible that a Schottky barrier or interfa-cial polarization effect is at work.The dielectric response of such a barrier on a semiconducting crystal is akin to that for the grain and grain boundary model.8It has also been ob-served that grains of CCTO have a high concentration of dislocation lines and twin boundaries that are likely present in the single crystal.5Such features can in principle lead to barrier layers that can contribute to the observed dielectric response of the crystal.Clearly more work on single-crystal CCTO is needed to better deﬁne the observed physics.B.The relaxational dynamics of CCTOExcept for the 3at.%Fe-doped sample,the low-T dielec-tric relaxations labeled ͓A ͔of the CCTO samples we exam-ined are not dipolar in nature,but arise from the resistance-capacitance ͑or RC ͒properties of the barrier layers.18Thus,they are not Debye relaxors,but exhibit Debye-like relax-ations with Arrhenius kinetics ͓1/␶=␻=␻0exp ͑−E /kT ͔͒,as observed by others.2,3A very signiﬁcant feature of the RC barrier-layer model is that the relaxation time deduced from the ␧Ј͑T ,␻͒data,␶␧Ј,is longer than that deduced from thetan ␦͑T ,␻͒data,␶␦.This is reﬂected in Figs.2and 3by ⌬T which is the difference in T between the inﬂection point in the ␧Јrelaxation,which determines ␶␧Ј,and the peak T in the tan ␦relaxation,which determines ␶␦.For a dipolar relax-ation process,␶␧Јϵ␶␦.The relatively large values of ⌬T in Figs.2and 3are clear indications of the nondipolar nature of the relaxation in CCTO.Figure 11͑a ͒and Table I summarize our results for relax-ation ͓A ͔.In the ﬁgure,T max is the temperature of the tan ␦peak at a given ␻.It is seen that the response of all the samples ͑except the 3at.%Fe sample ͒exhibit two Arrhenius segments,each with a distinct activation energy E .Thus,there are two separate activated conduction processes con-tributing to the relaxation labeled ͓A ͔.The data from all the samples are fairly tightly bunched together yielding essen-tially the same activation parameters E and ␻0͑=1/␶0͒,for all the samples.We have also plotted in Fig.11͑a ͒the sparse single-crystal data of Homes et al.2Although these authors drew a straight line through their data points,deducing E =54meV and ␻0=8.4ϫ10−9Hz,the results are clearly ei-ther non-Arrhenius or indicate two Arrhenius segments as suggested by the dashed lines through the solid triangles.The activation energy of the high-T segment is the same ͑ϳ110meV ͒as for our ceramic samples,but for the lower-T segment the suggested E is signiﬁcantly lower than the acti-vation energy for the ceramic CCTOsamples.FIG.10.Correlation between the value of ␧Јin the ␧Ј͑T ͒pla-teau region and the maximum value of tan ␦.The inset shows this correlation on a log-logplot.FIG.11.Arrhenius plots for the observed relaxations.͑a ͒Low-temperature relaxation ͓A ͔showing the two different thermally ac-tivated regions and the ͓A *͔relaxation for the 3at.%Fe-doped sample.͑b ͒shows the high-temperature relaxation for the 1and 3at.%Fe-doped samples.The inset in ͑b ͒shows the variation of the activation energy with Fe doping.GRUBBS et al.PHYSICAL REVIEW B 72,104111͑2005͒According to the grain and grain boundary model,these activation energies reﬂect the activation of conduction mechanisms in the grains.18In terms of magnitudes,the E’s for both segments are consistent with values for other semi-conducting titanate perovskites.4Sinclair et al.4obtained an activation energy for the bulk,or grains,of E=80meV from complex impedance measurements.Their data covered a nar-rower T range than our own,and consequently they observed only one Arrhenius segment.Their80meV value for the grainsﬁts in between our values for the two Arrhenius seg-ments.Given that the E for the higher-T segment for the single crystal agrees with the comparable E’s for the ceramics,we suggest that it represents the activation of the primary grain conduction mechanism in CCTO.As for the activation en-ergy of the lower-T Arrhenius segment,we note from Fig. 11͑a͒that the suggested E for the single crystal may be con-siderably lower than the E’s for the ceramic samples,but there are not sufﬁcient data points to be certain.This obser-vation,coupled with the fact that E in this regime for the ceramics is essentially independent of doping,suggests that we are dealing with an electronic trap of an impurity or de-fect commonly present in CCTO.A very likely candidate defect that may be responsible for the two observed activated segments in Fig.11͑a͒is the oxy-gen vacancy V O.This defect is undoubtedly introduced at a sufﬁciently high concentration during high-T crystal growth and ceramic processing.V O is known to be responsible for the semiconducting behavior of many oxide perovskites.4,19,20It acts as a double donor introducing two levels into the host’s band gap lying aboutϳ50–100and 100–200meV,respectively,below the conduction-band edge,depending on the material.These energies are sufﬁ-ciently close to our measured activation energies of Eϳ73 and108meV so as to make the role of V O in CCTO quite credible.The results in Fig.11͑a͒show that Nb doping and Fe doping at the1at.%level have no inﬂuence on the low-T conduction process in CCTO.However,the results for the3 at.%Fe-doped sample are very different.The response͓A*͔is that of a true dipolar relaxor with a single Debye relax-ation time and kinetic parameters of E=268meV and␻0 =2.12ϫ1013Hz.The nature of the dipolar entity responsible for this relaxation is not known.It could of course be due to an off-center substitutional Fe ion or an Fe-induced defect. As discussed in Sec.III A,the only reason we are able to resolve this relaxation for3at.%Fe is because,at this dop-ing level,the Fe reduces the dielectric loss͑conductivity͒and␧Јto such a large extent that the sample exhibits essen-tially its intrinsic dielectric response.It may well be that thisdipolar relaxation is present in all samples͑i.e.,it is not Ferelated͒,but it is not resolved because it is swamped by thehuge␧’s and tan␦peaks of the other CCTO samples.Our initial supposition and motivation for adding Fe wasthat Fe would substitute at the Ti4+͑B site͒producing holes. The results on the3%Fe sample and our magnetic data,to be presented in Sec.III D below,indicate that much of the Fe3+substitutes at the Ti4+site.The holes introduced by this Fe doping could then compensate for the conduction elec-trons contributed by the oxygen vacancies,leading to essen-tially intrinsic behavior of the CCTO grains.Our results sup-port this view.Another important feature in our results on the1and3at.%Fe-doped samples is the high-T relaxation labeled͓B͔in Figs.4and6.This Debye-like relaxation exhibits Arrhen-ius kinetics as shown in Fig.11͑b͒with the parameters listed in Table I.The rapid increase in tan␦with increasing T above350K in Fig.3͑also observed for undoped CCTO͒suggests that this relaxation may be obtained for the undoped and Nb-doped samples at TϾ400K͑beyond the range of the apparatus used͒.In terms of the barrier-layer model,this͓B͔relaxation is attributed to the grain-boundary region and thus the activa-tion energies in Table I and in Fig.11͑b͒reﬂect E for the conduction process in this region.This is the region that controls dc transport.The activation energies of E=542and 470meV for the1and3at.%Fe samples,respectively,are characteristic of activation energies for dc electronic trans-port in many oxides.Sinclair et al.4deduced a grain bound-ary activation energy of600meV for undoped CCTO fromTABLE I.Arrhenius kinetic parameters for various ceramic CCTO samples for two relaxation processes.See text for details.Sample E͑meV͒␻0͑s−1͒E͑meV͒␻0͑s−1͒Low-T Arrhenius segment͑a͒Relaxation͓A͔High-T Arrhenius segmentUndoped͑A͒71 1.8ϫ108104 1.0ϫ1010Undoped͑B͒71 2.6ϫ108106 1.3ϫ10101%Nb doped71 1.3ϫ1081109.7ϫ1094%Nb doped74 2.5ϫ1081099.9ϫ1091%Fe doped81 3.2ϫ1081138.8ϫ1093%Fe doped268 2.1ϫ1013͑b͒Relaxation͓B͔Undoped600a1%Fe doped542 6.8ϫ10123%Fe doped470 1.8ϫ1011From Sinclair et al.͑Ref.4͒determined from complex impedance data.DIELECTRIC AND MAGNETIC PROPERTIES OF Fe-…PHYSICAL REVIEW B72,104111͑2005͒。

磷酸铁锂和磷酸锰锂对比

I NSTITUTE OF P HYSICS P UBLISHING J OURNAL OF P HYSICS:C ONDENSED M ATTER J.Phys.:Condens.Matter16(2004)5531–5548PII:S0953-8984(04)79785-4A comparative study of magnetic properties of LiFePO4and LiMnPO4Denis Arˇc on1,2,Andrej Zorko1,Robert Dominko3and Zvonko Jagliˇc iˇc41Department of Solid State Physics,Institute Joˇz ef Stefan,Jamova39,1000Ljubljana,Slovenia2Faculty of Mathematics and Physics,University of Ljubljana,Jadranska19,1000Ljubljana,Slovenia3National Institute of Chemistry,Hajdrihova19,1000Ljubljana,Slovenia4Institute of Mathematics,Physics and Mechanics,Jadranska19,1000Ljubljana,SloveniaReceived23April2004Published16July2004Online at /JPhysCM/16/5531doi:10.1088/0953-8984/16/30/014AbstractA detailed comparative study of the magnetic properties of LiFePO4andLiMnPO4samples is presented.Magnetic susceptibility,electron paramagneticresonance and7Li NMR experiments were performed on samples as preparedfor electrochemical studies.The ground state of LiFePO4seems to be that ofa collinear antiferromagnet and very robust against crystal imperfections.Onthe other hand,our LiMnPO4samples possess a weak ferromagnetic groundstate with a transition temperature T N=42K.We suggest that solitons maybe very important magnetic excitations in these systems and that pinning ofsolitons below T N together with frustration plays a decisive role in the formationof the weak ferromagnetic state in LiMnPO4.The differences between themagnetic properties of these two samples reﬂect also the differences betweentheir electronic structures and may thus be important for the electrochemistryof LiFePO4and LiMnPO4.1.IntroductionSince the pioneering work of Goodenough et al[1],the phospho-olivines LiMPO4(where M stands for Fe,Mn,Co,Ni)have been recognized as a potential positive electrode material for use in lithium rechargeable batteries.This family of materials has numerous advantages over the layered rock salt oxides(e.g.LiCoO2and LiNiO2)that are currently used in commercial batteries.Besides its high capacity(∼170mA h g−1),LiMPO4cathode material shows high stability during lithium extraction/insertion and does not deteriorate when used at moderately high temperatures.Recently,a lot of effort has been devoted to optimization of LiFePO4 material.In particular,particle minimization with intimate carbon contact resulted in almost the theoretical capacity at moderate current densities with LiFePO4cathode material[2]. 0953-8984/04/305531+18$30.00©2004IOP Publishing Ltd Printed in the UK55315532D Arˇc on et al Surprisingly,only few papers indicate the applicability of LiMnPO4[3–5]and,besides that, the results remain controversial.The origin of the different electrochemical activities of isostructural LiFePO4and LiMnPO4is not quite clear at the moment.Yamada et al[3]suggested that the low electrochemical activity of LiMnPO4is due to slow kinetics and internal friction during lithium extraction/insertion.We have recently performed a comparative structural study[6]of these two materials and found minute differences in the oxygen vibrations in the Li layer.Yet another important factor is the magnetism in these compounds[7–9].Magnetic properties are determined by the electronic states and may thus reﬂect the potential differences within the LiMPO4family.Magnetism is therefore indirectly important for the electrochemical properties as well.In the olivine structure the magnetic ions occupy only the so-called M2site,i.e.the M2+ ion sits in the centre of the MO6unit.Each MO6distorted octahedron is connected to four other MO6via vertices forming a layered network perpendicular to the crystal[100]axis. Successive M2+layers are separated by PO4tetrahedra and LiO6octahedra.Magnetic properties of LiMPO4were studied in the early1960s.The magnetic structures of LiCoPO4and LiNiPO4are suggested to be those of collinear antiferromagnets[10]but they differ in the orientation of the sublattice magnetizations.In LiCoPO4the magnetic moments are oriented along the[010]direction while in LiNiPO4they point along the[001]direction. On the other hand,in LiFePO4the antiferromagnetic transition[10,11]occurs at52K with magnetic moments aligned along the crystal[010]axis.Andﬁnally,LiMnPO4was again suggested to be a collinear antiferromagnet with a N´e el temperature ranging between35K, for synthetic materials[8],and42K,for minerals[12].However,with the discovery of the electrochemical activity of LiFePO4there has been a revival of interest in these materials.In this paper we revisit the magnetic properties of LiFePO4and LiMnPO4materials,as prepared for use as an active material for Li ion batteries. We emphasize that the focus of this study is on the magnetic properties of materials prepared for electrochemical studies with all possible defects,vacancies and particle size effects.In particular,we have performed SQUID magnetization,EPR and7Li NMR measurements between room temperature and2K.The measurements revealed differences in the magnetic ground states that could be important for the electrochemical activity of the LiMPO4family.2.Experimental details2.1.Sample synthesisThe samples were prepared by a standard sol–gel method[13].As a starting precursor,iron (III)citrate(Aldrich,22,897-4)or manganese(II)acetate tetrahydrate(C4H6MnO4×4H2O, Fluka,63537)and citric acid(C6H8O7×H2O,Kemika,0319506)were dissolved at60◦C in water.Separately,an equimolar water solution of LiH2PO4was prepared from lithium phosphate(Li3PO4,Aldrich33,889-3)and phosphoric(V)acid(H3PO4,Aldrich31,027-1). Clear solutions were mixed together and dried at60◦C for24h.After thorough grinding with a mortar and pestle,the xerogel obtained wasﬁred in a reductive(5wt%of hydrogen in argon) atmosphere at700◦C(LiFePO4)or900◦C(LiMnPO4)for10h.The powder obtained was black due to carbon-coated submicron particles(the content of carbon is about6wt%)[14]. The average size of coherent domains obtained from the analysis of the half-width of the x-ray diffraction peaks was,for both materials,in the range between40and50nm.The average particle size did not vary substantially from batch to batch.A typical active surface area was estimated to be around60m2g−1.A comparative study of magnetic properties of LiFePO4and LiMnPO45533Figure1.Lithium extraction/insertion in thesecond cycle at a current density of8.5mA g−1(C/20rate).2.2.Electrochemical testingThe preparations of electrodes and systems for electrochemical testing are describedelsewhere[15].We should stress that the current density used for discharging/chargingwas set to a value of8.5mA g−1(corresponding roughly to a C/20rate).The cut-off voltages duringthe discharge and charge were4.1and2.7V for LiFePO4and4.5and2.5V versus the lithium metal reference for LiMnPO4.Typical charge and discharge curves for LiFePO4and LiMnPO4are shown inﬁgure1.Although the nominal capacities for the two materials are similar,the capacity obtained for LiMnPO4is three times lower than that for LiFePO4and,besides that,the large polarizationof the LiMnPO4sample suggests hindered lithium extraction/insertion.2.3.Magnetization and magnetic resonance measurementsThe susceptibility and magnetization were measured with a Quantum Design SQUIDmagnetometer,equipped with a5T superconducting magnet.Continuous wave(cw)EPR measurements were performed on a Bruker E580spectrometerusing a Varian dual resonator with a reference sample in the second resonator to account forthe slight changes in the Q-factor during the measurements.In all the cw experiments a modulationﬁeld H mod=1G andνmod=100kHz were used.The temperature was stabilized within0.2K in a continuousﬂow cryostat,ESR900.To estimate the EPR spin susceptibility,Cu(SO4)2·5H2O has been used as a standard sample.7Li NMR measurements were performed at a Larmor frequencyνL=104.9MHz in a superconducting magnet,at6.3T.The7Li NMR spectra were,over the entire temperature range,broader than the measurement window determined with the90◦rf pulse width6µs. The spectra were therefore measured via the intensity of the echo signal while sweeping the resonance frequency in steps of10kHz.We also used automatic tuning of the resonance circuit in the frequency range between103.5and106MHz.The spin–lattice relaxation time T1was measured using a saturation–recovery technique with appropriate phase cycling.3.Results3.1.Magnetic susceptibility measurementsInﬁgure2we show the temperature dependence of the static spin susceptibility in LiMnPO4 powder.We stress that we found in this particular sample traces of Li3PO4impurities.In5534D Arˇc on et alFigure2.The temperature dependence of the(a)spin susceptibility and(b)inverse spinsusceptibility for LiMnPO4powder.Please note the difference between theﬁeld cooled(FC)and zero-ﬁeld cooled(ZFC)experiments.an independent check,we tried to examine the magnetic properties of Li3PO4using EPR butfailed toﬁnd any paramagnetic signal.We therefore conclude that the Li3PO4impurities cannot contribute to the total spin susceptibility presented inﬁgure2.At high temperatures,i.e.at temperatures between room temperature and50K,we observe a Curie–Weiss behaviour witha Curie constant C=4.23(1)emu K mol−1and a Curie–Weiss temperature =−87(2)K. These constants do not vary signiﬁcantly from sample to sample.The susceptibility also shows the same behaviour between room temperature and∼40K,irrespective of the coolingprotocol.However,around40K we observe an anomaly which is strongly dependent on whether the sample has been cooled in zeroﬁeld or in a magneticﬁeld.While in zero-ﬁeldcooled experiments we observe a cusp in the magnetic susceptibility at T N=39–42K,in ﬁeld cooled experiments the susceptibility suddenly increases below T N and then at lower temperatures saturates.The same type of behaviour can also be seen from the temperaturedependence of the inverse susceptibility(ﬁgure2(b))where1/χsuddenly decreases at T N. We stress that the magnitude of the saturated moment slightly varies from sample to sample. However,it will be shown later with7Li NMR measurements that this is still an intrinsic phenomenon and not a result from some unknown impurity phase.Further evidence for the magnetic ordering comes from the ac susceptibility measurements,which clearly show a peak at T N.We would like to stress that the peak position is frequency independent,i.e.it is precisely the same atν=1Hz as atν=100Hz,excluding the possibility of some spin-glass or superparamagnetic-like behaviour.A comparative study of magnetic properties of LiFePO4and LiMnPO45535Figure3.Magnetization curves measured for powdered LiMnPO4at50K(open circles)and at2K(solid circles).The measurements are displayed only forﬁelds between−6and6kOe,whilecomplete magnetization versus magneticﬁeld cycles are shown in the inset.The arrows indicatethe direction of the change of the magneticﬁeld.Finally,we show inﬁgure3hysteresis curves measured for a LiMnPO4powder sample. One can clearly see the hysteresis in the2K curve.The coerciveﬁeld is about1000Oe,whilethe remanent magnetization is about70emu mol−1.This remanent magnetization corresponds to0.012µB/Mn.Interestingly,weﬁnd a structure in the hysteresis just around zeroﬁeld.The hysteresis disappears at T=50K,i.e.at temperatures above T N.At this temperature wemeasure only a straight paramagnetic line.The magnetic susceptibility of LiFePO4is less complicated(ﬁgure4).Again,at hightemperatures the magnetic susceptibility follows a Curie–Wess behaviour with a Curie constant C=4.28emu K mol−1and a Curie–Weiss temperature =−115(1)K.The susceptibility clearly shows a maximum characteristic of antiferromagnetic ordering at50K.No noticeable difference between zero-ﬁeld cooled andﬁeld cooled experiments has been found.On the basis of these experiments,we can conclude that LiFePO4goes into a collinear antiferromagnetic ground state below T N=51(1)K.3.2.EPR measurementsThe EPR line in LiMnPO4has a Lorentzian lineshape over the entire temperature range.This result is somewhat surprising as typically for a2D lattice formed by the Mn ions one would expect deviations from the Lorentzian lineshape as a result of the spin diffusion processes. The EPR peak-to-peak linewidth is,at room temperature,298G.We note that this linewidth is larger by an order of magnitude than the one measured for perovskite layered Mn square-lattice antiferromagnets[16].The temperature dependence of the EPR signal intensity,which is,in the paramagnetic phase,directly proportional to the static spin susceptibility,is shown inﬁgure5(a).At room temperature the intensity of the line corresponds to the spin susceptibilityχEPR= 1.05×10−2emu mol−1,which is nearly the same as the one measured by the SQUID(ﬁgure2). This proves that we detect all Mn spins in these EPR measurements.At high temperatures the intensity of the signal follows the one measured by the SQUID magnetometer.The temperature dependence of the EPR spin susceptibility can beﬁtted with a Curie–Weiss law:,(1)χEPR=CT−5536D Arˇc on etal Figure4.The temperature dependence of the(a)spin susceptibility and(b)inverse spinsusceptibility for LiFePO4powder.where C=3.92(2)emu K mol−1is the Curie constant and =−105(2)K is the Curie–Weiss temperature.We note that the extracted Curie–Weiss temperature is slightly more negative than the one obtained from SQUID measurements.On cooling below around45K the EPR signal starts to disappear quite rapidly;it nearly vanishes in an interval of a few kelvins and almost completely vanishes at42K.This proves thatthe magnetic transition observed in SQUID measurements is indeed intrinsic and is not related to some unidentiﬁed impurities.We were not able to detect antiferromagnetic resonance in powdered LiMnPO4below T N.We have however been able to measure the residual EPR signaldown to4K.The origin of this signal will be discussed later.Although the centre of the line(i.e.the g-factor)is nearly temperature independent (ﬁgure5(b)),the linewidth exhibits a pronounced temperature dependence(ﬁgure5(c)).Inthe entire temperature interval between room temperature and T N the linewidth increases monotonically with decreasing temperature.At T N the linewidth nearly diverges.Inﬁgure6we show the temperature dependence of the EPR linewidth on two different scales:ln H versus(T−T N)/T N and ln( H− H(∞))versus J/T.In theﬁrst graph we are testing the power-law behaviour while in the second one we are looking for an activated type of behaviourof the EPR linewidth.An attempt toﬁt the temperature dependence of the EPR linewidth tothe power-law behaviourH(T)= H0+ H P0T−T NT N−β(2)A comparative study of magnetic properties of LiFePO4and LiMnPO45537 Figure5.The temperature dependence of the(a)intensity,(b)centre and(c)linewidth of theLorentzian EPR signal for LiMnPO4powder.(Thisﬁgure is in colour only in the electronic version)is shown inﬁgure6(a).The parameters used in theﬁt are the linewidth at inﬁnite temperature H0=236(2)G, H P0=134(3)G and the power exponentβ=0.45(1).Surprisingly,this ﬁt successfully describes the temperature dependence of the linewidth over a wide temperature interval,although it is difﬁcult to believe that the criticalﬂuctuations would survive even at temperatures as high as10T N!On the other hand,we notice that one can describe the temperature dependence of the EPR linewidth in the paramagnetic phase also with an activated type behaviour H∝exp(E a/T) where E a=120K(see the dotted curve inﬁgure6(b)).In that case the deviation of the temperature dependence of the EPR linewidth from the activated type of behaviour just above the T N reﬂects the criticalﬂuctuations.The data were thusﬁtted with a sum of two contributions, i.e.the activated type contribution and the power-law contributionH(T)= H0+ H a0exp(E a/T)+ H P0T−T NT N−β.(3)An unconstrainedﬁt(the solid curve inﬁgure6(b))led to the activation energy E a=144(3)K and power exponentβ=0.38(6).A deviation at higher temperatures(i.e.at lower J/T values)5538D Arˇc on et alFigure6.The temperature dependence of the EPR linewidth for LiMnPO4powder in(a)a‘power-law’graph and(b)an‘activation-law’graph.can also be a signature of the spin diffusion contribution to the EPR linewidth,which makes H0weakly temperature dependent.The possible origin of the activated type behaviour of the EPR linewidth will be discussed later.Finally,we would like to mention that we were not able to detect the X-band EPR signal for LiFePO4,most probably due to the extreme linewidth of the signal.3.3.7Li NMR spectraInﬁgure7we compare the temperature dependences of the7Li NMR signals measured for LiMnPO4and LiFePO4.We stress that the linewidth exceeds our excitation window even at room temperature and that we were thus forced to measure the spectra by sweeping the resonance frequency.We also note that the lineshape does not simply correspond to quadrupole broadened powder spectra.The lineshape has a shoulder on the high frequency side of the spectrum and can be satisfactorilyﬁtted to the lineshape expected for a uniaxial shift tensorinteraction.This would suggest that we are in fact observing only a central−12↔12transitionbroadened due to a coupling to electronic spins residing at the metal sites.The satellite transitions may be smeared out over the whole frequency range and they are very difﬁcult to detect in powder samples.We also note that the shift from the Larmor frequency is rather small, suggesting that the contact hyperﬁne coupling constants are rather small for these compounds. This result is in agreement with recent high temperature7Li MAS NMR results[23].A comparative study of magnetic properties of LiFePO 4and LiMnPO 45539(a)(b)Figure 7.The temperature dependence of the 7Li NMR signal for (a)LiMnPO 4and (b)LiFePO 4powder.On cooling,the 7Li NMR line is shifted to lower resonance frequencies and continuously broadens for both samples.This reﬂects the Curie–Weiss temperature dependence of the magnetic susceptibility.However,below the N´e el temperature the two samples behave qualitatively differently.While for the LiMnPO 4sample the linewidth tends to increase even below T N ,for LiFePO 4the line narrows.This difference indicates that different local ﬁelds are experienced by the 7Li nuclei below T N in the particular two samples studied.The results clearly suggest that in LiFePO 4the magnetic ordering is indeed antiferromagnetic.This is seen from the fact that the local ﬁeld at the 7Li site decreases below the transition temperature.On the other hand,in LiMnPO 4the magnetic ordering is weak ferromagnetic type as evidenced by the broadening of the 7Li NMR spectra.Possible origins of the weak ferromagnetism in our LiMnPO 4sample will be discussed later.The changes in the lineshape can also be seen in the temperature dependence of the moments of the 7Li NMR spectra (ﬁgure 8).The temperature dependence of √M 2,which is approximately a measure of the linewidth of the spectra,on cooling steadily increases for both samples (ﬁgure 8(a)).For instance,√M 2increases from 82kHz at room temperature to around 200kHz at 60K.However,below the N´e el temperatures again the two samples behave qualitatively differently.√M 2for LiMnPO 4suddenly increases below 40K and then nearlysaturates at the value of 350kHz.On the other hand,for LiFePO 4the second moment below 50K starts to decrease and,for instance,at 15K it amounts to only √M 2∼135kHz.The same observations seem to hold for the ﬁrst moments as well (ﬁgure 8(b)).For both samplesthe line ﬁrst paramagnetically shifts on cooling.However,at T N the line continues to shift for LiMnPO 4while for LiFePO 4the shift of the 7Li NMR line reverses its trend.So,to summarize this section,the local ﬁelds experienced by the 7Li NMR probe on cooling from room temperature at ﬁrst increase as expected for a paramagnetic phase.However,below T N the two samples are qualitatively different.In LiFePO 4the local ﬁeld decreases while in LiMnPO 4the local ﬁeld increases below T N .This suggests that the ground state of the sample of LiFePO 4studied is indeed antiferromagnetic,but that of LiMnPO 4is weak ferromagnetic.3.4.7Li relaxationSpin dynamics is reﬂected in the spin–spin relaxation time T 2and spin–lattice relaxation time T 1.The spin–spin relaxation time has been measured in two different ways:ﬁrst by5540D Arˇc on et alFigure 8.The temperature dependences of the (a)second-moment and (b)the ﬁrst-moment 7LiNMR spectra of LiMnPO 4(solid circles)and LiFePO 4(open circles).monitoring the decay of the Hahn echo as a function of the separation time τbetween the two pulses and second by means of the Carr–Purcell–Meiboom–Gill (CPMG)sequence [24].Typical relaxation curves measured for LiMnPO 4are shown in ﬁgure 9.The magnetization curve obtained from the decay of the Hahn echo is clearly non-exponential.A non-exponential decay is usually a signature of diffusion processes and has a characteristic time dependence:E HE (τ)=E 0exp −2τT 2−23(γG )2D τ3 .(4)Here T 2is the decay rate of transverse magnetization in a uniform ﬁeld,τis the time interval between the two pulses in the Hahn-echo experiment,γis the nuclear gyromagnetic ratio,G =∂B /∂z is the gradient of the magnetic ﬁeld,which is assumed to be uniform,and D is the appropriate self-diffusion coefﬁcient.In many cases one wishes to determine the diffusion constant by intentionally applying a ﬁeld gradient.However,even if the external magnetic ﬁeld is homogeneous,local inhomogeneity in the spin susceptibility may take over and effectively lead to a very similar behaviour.In such a case one would expect to ﬁnd G =∂χ∂z B ,which could thus be relevant in the paramagnetic systems.To avoid the extra decay due to diffusion,the Carr–Purcell–Meiboom–Gill (CPMG)sequence is often used.The experiments (a typical example is shown on ﬁgure 9(b))were performed using different times τbetween 180◦pulses.The data obtained were then analysed using a simple exponential form E (τ)=E 0exp −2τT 2 .We note however that the extracted spin–spin relaxation time T 2depends on the value of τused in the CPMG sequence.A typical(a)(b)Figure9.The decay of the echo intensity in LiMnPO4as a function ofτin(a)a simple Hahn-echoexperiment and(b)a Carr–Purcell–Meiboom–Gill(CPMG)experiment.In the inset we show thedecay of the echo in the CPMG experiment.The temperature is296K here.Figure10.The dependence of the relaxation parameter T2obtained from the CPMG measurementsfor(a)LiFePO4and(b)LiMnPO4at T=290K.example of such a dependence is shown inﬁgure10.For very small values ofτ,i.e.when τ=30µs,the extracted parameter T2approaches the values obtained from a two-pulse Hahn-echo experiment.For larger values ofτ>50µs,the extracted parameter T2on the other hand becomes smaller by almost a factor of two.The effect is much more pronounced for LiFePO4, indicating that the slow motions responsible for theτdependence of the parameter T2are more active on the timescale ofτ∼30–50µs.Inﬁgure11we show the temperature dependence of the spin–spin relaxation time as obtained from the CPMG sequence withτ=30µs for both samples.Interestingly,the relaxation time is,around room temperature,more or less temperature independent and,for instance,for LiFePO4amounts to590±25µs,while for LiMnPO4it is slightly smaller and corresponds to180±35µs.On cooling however,T2exhibits anomalous behaviour.For instance,for LiFePO4,about10K above the transition temperature,T2starts to get smaller,Figure11.The temperature dependence of the spin–spin relaxation time T2obtained with the two-pulse Hahn-echo method and the CPMG method(hereτ=30µs).i.e.the relaxation rate T−12becomes enhanced.At47K we thus have T2=340±12µs,whichis smaller by a factor of1.7than the room temperature value.Below the antiferromagnetictransition temperature,T2starts to rapidly increase with decreasing temperature and reaches 3.49±0.05ms at12K.Such behaviour signals the antiferromagnetic criticalﬂuctuations justabove the transition temperature and then,in the magnetically ordered phase,a slowing downof the electron spin dynamics as expected.Interestingly we do notﬁnd clear evidence for critical behaviour of LiMnPO4,but below the N´e el temperature the spin–spin relaxation timeincreases again.We have also measured the spin–lattice relaxation time using a saturation–recovery method.The relaxation curves were found to be of the stretched exponential formM Z(t)=M0(1−r exp(−(t/T1)α))(5) over the entire temperature range.This suggests that the dominant relaxation mechanism is the ﬂuctuation of the hyperﬁne coupling to the Mn2+magnetic moments.Furthermore,we also notethat the spin–lattice relaxation times T1obtained are very small—over most of the temperaturerange,of the order of1–2µs,again requiring electron–nuclear hyperﬁne coupling.Inﬁgure12 we show the temperature dependence of the parameter T1.The stretched exponential factorαwas found to be very weakly temperature dependent.We see that the parameter T1is more or less temperature independent between room temperature and60–70K and,for instance,forLiMnPO4corresponds to about1.5ms.Just above the transition temperature the spin–latticerelaxation rate1/T1becomes enhanced(ﬁgure12).Such an enhancement of the relaxation rate signals a critical slowing down of the magneticﬂuctuations,as we have seen in T2measurements.This effect is again more obvious for LiFePO4than LiMnPO4.Below T Nthe spin–lattice relaxation time T1suddenly starts to increase(see the inset toﬁgure12), signalling the freezing out of the electronic spin dynamics.We also note that the parameter αis for LiMnPO4about0.85±0.1and for LiFePO4is about0.96±0.05.Such stretched exponential forms of the magnetization relaxation curves are often found in systems exhibitingfrustration and/or local disorder.A simple conclusion would thus be that the local disorderand frustration play more important roles for LiMnPO4than for LiFePO4.Figure12.The temperature dependence of the spin–lattice relaxation rate1/T1for LiMnPO4(open circles)and LiFePO4(solid circles).In the inset we show the temperature dependence ofthe spin–lattice relaxation time T1for LiFePO4.4.DiscussionThe structures of LiMnPO4and LiFePO4are isomorphous.They both crystallize in the olivine type crystal structure with orthogonal space group Pnma.The proﬁle reﬁnement by the FULLPROF program gives for LiMnPO4the room temperature lattice constants a=10.4447(6)Å,b=6.1018(3)Å,c=4.7431(3)Å[6].The Mn–Mn distance is shortest in the b–c plane,at3.92Å.The Mn–Mn distance is much longer in the perpendicular direction, along the crystal a-lattice,i.e.5.62Å.These parameters suggest a quasi-two-dimensional character for the magnetic properties of LiMnPO4.However,the Mn–O–Mn bonding is rather complicated and the Mn lattice cannot be simply treated as a two-dimensional square lattice. This is clearly reﬂected in the temperature dependence of the magnetic susceptibility,which can be,in the paramagnetic phase,simplyﬁtted with a Curie–Weiss law(equation(1)).An unconstrainedﬁt of the temperature dependence of the LiMnPO4susceptibility leads to a Curie temperature =−87±2K and a Curie constant C=4.23(1)emu K mol−1.The Curie constant obtained corresponds to the effective magnetic momentµeff=5.82(2)µB,which is close to but slightly less than the value expected for Mn(II)spins(S=5/2).A similar observation holds also for LiFePO4,where the susceptibility again followsa Curie–Weiss law between room temperature and60K with a Curie constant C=4.28(2)emu K mol−1and a Curie–Weiss temperature =−115(1)K.In this case the Curie constant corresponds to the effective momentµeff=5.85(4)µB,which is in agreement with the Fe2+high spin S=5/2state.We note that the temperature dependence of the susceptibility does not follow the dependence predicted for the2D square lattice of classical spins[17]as it does not show a typical broad maximum.The sign of the Curie temperature for both samples suggests predominantly antiferromagnetic coupling between the spins.We stress once again that we obtained,in EPR measurements,a spin-only susceptibility for about15%higher Curie temperatures(see section3.2).One can use a classical result to estimate the exchange coupling constant:J=3k B2zS(S+1)(6)where z is the number of nearest neighbours.The estimated exchange coupling constant is J ex=2.5K for LiMnPO4,while for LiFePO4it is J ex=3.3K.。

Energy gap and London penetration depth of MgB_2 films determined by microwave resonator me

Submitted to Physical Review Letters,ondon penetration depth of MgB2 films determined by microwave resonator measurements N. Klein, B.B. Jin, J. Schubert, M. Schuster, H.R. Yi Forschungszentrum Jülich, Institute of Thin Films and Interfaces, D-52425 Jülich, Germany, A. Pimenov, A. Loidl Universität Augsburg, Experimentalphysik V, EKM, 86135 Augsburg, Germany S.I. Krasnosvobodtsev P.N. Lebedev Physics Institute, Russian Academy of Sciences, 117924 Moscow, Russia
Eq.1
with G = 740 Ω being a geometrical factor determined by numerical simulation of the electromagnetic field distribution in the cavity and Qniobium( 4.2 K) = 92000 representing the unloaded qualtity factor measured by employing a high-quality niobium thin film as sample. For temperature below 30 K Eq. 1 allows for the determination of Rs with a systematic error of about 0.1 mΩ, which is due to neglecting the temperature dependent background losses of the cavity and the small microwave losses (Rs ≈ 10-5Ω) of the niobium film. The temperature dependence of the penetration depth was determined from the temperature dependence of the resonant frequency f(T) using G f (T ) − f (4.2 K ) πµ 0 f 2 (4.2 K )

Temperature Dependence of the Spin Polarization of a Quantum Hall Ferromagnet

1
A rapidly growing body of evidence, both theoretical [1{3] and experimental [4{7], strongly suggests that the lowest-lying charged excitation of the spin-polarized = 1 quantum Hall state is a spin-texture called a Skyrmion. This many-body state consists of radial spin density that is reversed at the center but gradually heals to the spin background over many magnetic lengths. The spin density distribution is determined by the interplay of the ferromagnetic exchange interaction and the Zeeman energy. The exchange interaction favors large Skyrmions while the Zeeman term favors smaller excitations. In GaAs samples presently under investigation the exchange energy can be up to two orders of magnitude larger than the Zeeman energy. Hartree-Fock calculations [2] predict that Skyrmions should consist of 3 4 spin ips per unpaired ux quantum for small excursions about = 1, a result consistent with early experimental work on the lling factor ( ) dependence of the zero temperature spin polarization near = 1 [4,6]. This dominance of the exchange interaction over the Zeeman energy has led theorists to refer to the = 1 quantum Hall state as the quantum Hall ferromagnet. In GaAs heterostructures experimentalists are presented an unprecedented opportunity to probe the physics of two-dimensional electron ferromagnetism in a well-characterized system. Thus insights gained from the thermodynamics of the spin polarization will be of interest not only to those studying many-body e ects in the integral quantum Hall regime, but also more generally, may elucidate the physics of 2D electron magnetism. In this communication we report on the experimental determination of the spin polarization as a function of temperature for such a = 1 quantum Hall ferromagnet. The system consists of a single-side n-modulation doped AlGaAs-GaAs single quantum well (SQW). The well thickness is 250 with an electron density of Ns = 1:8 10 cm and mobilA ity = 2:6 10 cm =V s. In order to perform absorption measurements the samples were mounted strain-free and thinned to 0:5m. The spin polarization is monitored through band-to-band absorption spectroscopy which distinguishes the occupancy of the two electron spin states. Band-gap absorption spectra show striking temperature dependence due to changes in the occupations of the spin-split states of the ground Landau level at lling factors

A New Equation of State for Carbon Dioxide Covering the Fluid Region

1514
1514 1516 1516 1516 1516 1517 1517 1518 1518
3.3 3.4 3.5 3.6 3.7 3.8 4.
Melting Pressure. . . . . . . . . . . . . . . . . . . . . . .. Sublimation Pressure. . . . . . . . . . . . . . . . . . . .. Vapor Pressure ....................... " Saturated Liquid Density. . . . . . . . . . . . . . . .. Saturated Vapor Density. . . . . . . . . . . . . . . . .. Caloric Data on the Liquid-Vapor Phase Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Experimental Basis of the New Equation of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.1 Thermal Properties .................... " 4.2 Specific Isobaric Heat Capacity. . . . . . . . . . .. 4.2.1 Experimental Results for the Specific Isobaric Heat Capacity. . . . . . . . . . . . .. 4.2.2 Results for the Specific Isobaric Heat

Temperature dependence of density profiles for a cloud of non-interacting fermions moving i

I. INTRODUCTION
The achievement of Bose-Einstein condensation in ultracold gases of bosonic atoms has given great impulse to the study of dilute quantum gases inside magnetic or optical traps [1]. Similar cooling and trapping techniques are being used to drive gases of fermionic alkali atoms into the quantum degeneracy regime [2]. It is also possible to experimentally generate and study strongly anisotropic atomic ﬂuids eﬀectively approaching dimensionality D = 2 or D = 1, at very low temperature and with very high purity, inside magnetic traps where the transverse and the longitudinal conﬁnement are vastly diﬀerent [3]. In the relevant conditions of temperature and dilution the atomic interactions become negligible in a gas of fermions placed in a single Zeeman sublevel inside a magnetic trap [4]. One can obtain in this way a close laboratory realization of an inhomogeneous, non-interacting Fermi gas, which has special signiﬁcance in regard to the foundations of density functional theory and to tests of the Thomas-Fermi approximation [5]. In the quasi-onedimensional (1D) case the ground state and the excitation spectrum of such an ideal gas of spin-polarized (or ”spinless”) fermions can be mapped into those of a gas of hard-core impenetrable bosons [6]. The latter is known as the Tonks gas, from the early work of Tonks [7] on the equation of state of hard-object adsorbates. Advances in atom waveguide technology, with potential applications to atom interferometry and integrated atom optics, especially motivate theoretical studies of dilute gases in a regime where the dynamics becomes essentially 1D [8]. The wave functions of a cloud of spinless fermions under harmonic conﬁnement are, of course, Slater determinants of harmonic-oscillator single-particle orbitals. Their representation in terms of Hermite polynomials has, however, limited usefulness for numerical calculations on mesoscopic clouds [9]. Brack and van Zyl [10] have developed a more powerful method for non-interacting fermions occupying a set of closed shells under isotropic harmonic conﬁnement in D dimensions, leading to analytical expressions for the particle and kinetic energy densities at temperature T = 0 in terms of Laguerre polynomials. These expressions are especially useful for numerical calculations on isotropic systems in D = 2 and D = 3. A Green’s function method, which altogether avoids the use of wave functions in favor of the matrix elements of the position and momentum operators, has been developed for similar purposes [11] and extended to calculations of the pair distribution function in the ground state for the 1D ideal Fermi gas [12]. This method has also been extended to the ground state of ideal Fermi gases under harmonic conﬁnement of arbitrary anisotropy in higher dimensionalities [13]. The purpose of the present paper is to extend the Green’s function method to an inhomogeneous 1D cloud of noninteracting fermions at ﬁnite temperature and to illustrate its usefulness by numerical calculations of the particle and kinetic energy density proﬁles as functions of temperature in the case of harmonic conﬁnement. Analytical expressions have been derived by Wang [14] for the same system at high temperature, where the chemical potential is lower than the single-particle ground-state energy. The case of non-interacting fermions under 3D harmonic conﬁnement has been treated by Schneider and Wallis [15] through the use of Laguerre polynomials. The emphasis of our numerical calculations will be to illustrate how the characteristic quantum features of the fermion cloud, i.e. its shell structure and the spill-out of particles at the boundaries beyond the Thomas-Fermi radius, evolve with increasing temperature as a semiclassical regime is being approached.

Bulk Viscous Cosmology

a rXiv:as tr o-ph/961189v131Jan1996Bulk Viscous CosmologyWinfried Zimdahl ∗Departament de F´ısica,Universitat Aut`o noma de Barcelona E-08193Bellaterra (Barcelona),Spain February 5,2008Abstract The full causal M¨u ller-Israel-Stewart (MIS)theory of dissipative processes in relativistic ﬂuids is applied to a ﬂat,homogeneous and isotropic universe with bulk viscosity.It is clariﬁed in which sense the so called truncated version is a reasonable limiting case of the full theory.The possibility of bulk viscosity driven inﬂationary solutions of the full theory is discussed.As long as the particle number is con-served almost all these solutions exhibit an exponential increase of the temperature.Assuming that the bulk viscous pressure of the MIS theory may also be interpreted as an eﬀective description for particle production processes,the thermodynamical behaviour of the Universe changes considerably.In the latter case the temperature increases at a lower rate or may remain constant during a hypothetical de Sitter stage,accompanied by a substantial growth of the comoving entropy.PACS numbers:98.80.Hw,04.40.Nr,95.30.Tg,05.70.Ln1IntroductionNonequilibrium thermodynamical processes are supposed to play a crucial role in the physics of the early Universe.Traditionally,for the description of these phenomena the theories of Eckart[1]and Landau and Lifshitz[2] were used.Due to the work of M¨u ller[3],Israel[4],Israel and Stewart[5, 6],Pav´o n,Jou and Casas-V´a zquez[7],Hiscock and Lindblom[8]it became clear however,that the Eckart type theories suﬀer from serious drawbacks concerning causality and stability.These diﬃculties could be traced back to their restriction toﬁrst order deviations from equilibrium.If one includes higher order deviations as well,the corresponding problems disappear.By now it is generally agreed that any analysis of dissipative phenomena in rela-tivity should be based on the theories of M¨u ller,Israel and Stewart,including at least second order deviations from equilibrium,although in speciﬁc cases the latter might reproduce results of the Eckart theory[9].Cosmological implications of second order theories wereﬁrst considered by Belinskii et al.[10].In the realm of cosmology especially bulk viscous phenomena have at-tracted considerable interest(see,e.g.,[11])since bulk viscosity is the only possible dissipative mechanism in homogeneous and isotropic spacetimes. While the coeﬃcient of bulk viscosity vanishes both for pure relativistic and pure nonrelativistic equations of state,it may be important,e.g.,for mix-tures of radiation and matter[12].On the other hand,it is well known [13-15]and widely used[17-25]that particle production processes in the ex-panding Universe may be phenomenologically described in terms of eﬀective viscous pressures.A major point of interest in the study of bulk viscous universes has been the question whether there are conditions under which a suﬃciently large bulk viscous pressures could lead to an inﬂationary behaviour.While some authors concluded that a bulk viscosity driven inﬂation is impossible [26],others[27-29]found inﬂationary solutions of the cosmological evolution equations.Partially,these diﬀerences occur since diﬀerent equations of state were used.But the results also depend on whether the investigations were performed within the full,causal second order theory or in a truncated version of the latter.One should be aware that discussing the issue of bulk viscous driven inﬂationary solutions at all,implies in any case an extrapolation of nonequi-librium thermodynamical theories beyond the range for which their appli-cability was strictly justiﬁed[29].Bulk viscous inﬂation,if it exists,is a far-from-equilibrium phenomenon,while even the full causal second order2MIS theory is a theory for small deviations from equilibrium.Therefore all theoretical conclusions are necessarily tentative.It is the hope that they nevertheless will provide an indication of the correct behaviour far from equilibrium.As to the relation between the full theory and the truncated version,to be discussed in some detail below,the following comment should be made from the outset.To decide whether a theory is truncated or not on apparently obvious formal grounds,i.e.,from the appearance of the causal evolution equations may be misleading.The structure of the evolution equation de-pends on the choice of the basic thermodynamical variables.In most cases the latter are equilibrium variables and the Gibbs equation has their fa-miliar form.In the framework of‘Extended Irreversible Thermodynamics’(EIT)however,a generalized Gibbs equation is used which includes dissipa-tive quantities as independent variables.Temperature and pressure in EIT are nonequilibrium quantities,diﬀerent from their equilibrium counterparts. Written in terms of these nonequilibrium quantities the causal evolution equation formally may look identical to the truncated theory written in terms of the more familiar equilibrium variables.These points have been clariﬁed in a recent paper by Gariel and Le Denmat[30]who pointed out that the apparently(because of the use of nonequilibrium variables)trun-cated theory of Pav´o n et al.[7]in fact is equivalent to the full theory of Israel[4].The reader is also referred to corresponding comments in[28]and [29].In the present paper the symbols T and p always denote the equilibrium temperature and the equilibrium pressure,respectively.In section2we reconsider the bulk viscous cosmological dynamics within the causal second order theories.The evolution law for the temperature of the cosmicﬂuid is shown to be diﬀerent in general from the ad hoc relation-ship used in previous treatments by Romano and Pav´o n[31],Zakari and Jou [28]and Maartens[29](subsection2.1).In subsection2.2we discuss the con-ditions under which the truncated version yields results close or identical to those of the full theory.The general viscousﬂuid dynamics of the full MIS theory is presented in subsection2.3.Subsection2.4investigates the con-ditions for viscous exponential inﬂation.Almost all corresponding solutions imply an exponential increase of theﬂuid temperature.As a consequence of this behaviour there is in general no substantial growth of the comov-ing entropy as was found previously[29].The latter result corresponds to a very speciﬁc limiting case.In section3the viscous pressure of the full causal theory is assumed to describe partially or fully the eﬀect of particle3creation taken into account by a nonvanishing source term in the particle number balance.In this setting the backreaction of the viscous pressure on the temperature is diﬀerent from the conventional viscousﬂuid case of section2.In the limit that the viscous pressure is entirely due to particle production there exist stable inﬂationary solutions for which both the par-ticle number density and the temperature are constant and,moreover,the comoving entropy grows exponentially.Section4summarizes the results of the paper.Units have been chosen so that c=k B=1.2Bulk viscousﬂuid dynamics2.1General relationsThe energy momentum tensor of a relativisticﬂuid with bulk viscosity as the only dissipative phenomenon isT ik=ρu i u k+(p+π)h ik.(1)ρis the energy density,u i is the4-velocity,p is the equilibrium pressure,h ik is the projection tensor h ik=g ik+u i u k,andπis the bulk viscous pressure. The particleﬂow vector N a is given byN a=nu a,(2)where n is the particle number density.Limiting ourselves to second order deviations from equilibrium,the entropyﬂow vector S a takes the form[5, 26]S a=sN a−τπ2respectively,whereΘ≡ua;ais theﬂuid expansion and˙n≡n,a u a -bining(6)and(7)with the Gibbs relationT ds=d ρn,(8)we getnT˙s=−Θπ.(9) From(3)and(6)weﬁndS a;a=−πζ˙π+1ζTu a ;a (10)for the entropy production density S a;a.The simplest way to guarantee S a;a≥0implies the evolution equationπ+τ˙π=−ζΘ−1τ−˙ζT (11)forπ,leading toS a;a=π2τ−˙ζT=0(14)is fulﬁlled,the full and the truncated theories become identical.As we shall see this is possible only in exceptional cases.Let us assume equations of state in the general formp=p(n,T)(15)5andρ=ρ(n,T),(16) according to which the particle number density n and the temperature T are our basic thermodynamical variables.Diﬀerentiating the latter relation, using the balances(6)and(7)as well as the general thermodynamic relation∂ρn−T∂T,(17)oneﬁnds the following evolution law for the temperature(cf.[22]):˙T=Θn∂ρ/∂n∂ρ/∂T,(18) or˙T∂ρ/∂T +π2nT,p=nT and T≪m we recover T m∼R−2in the matter dominated case.For a viscousﬂuid the behaviour of the tem-perature depends onπ.Sinceπis expected to be negative,the second term in the bracket on the r.h.s.of(19)will counteract theﬁrst one.Close to equilibrium,i.e.,for|π|<p the existence of a bulk viscous pressure implies that in an expanding universe the temperature decreases less rapidly than in the perfectﬂuid case.2.2The truncated versionWhile the truncated version was used in most of the earlier applications, more recently an increasing number of authors[26,28,31,29]has studied the full theory and compared the results of the latter wih those of the trun-cated version.In some cases these results diﬀer dramatically,which may be interpreted as a breakdown of the Maxwell-Cattaneo type equations as a reasonable approximation to the full theory under the corresponding con-ditions.What seems to be missing,however,are general criteria according to which one may decide whether the truncated version is sensible and be-yond which limits it fails to give an answer close to that of the full theory. Intuitively one expects the coincidence to be the better the closer one is to6the equilibrium case.Below we shall give an example showing that there are identical results in exceptional cases even far from equilibrium.In order to clarify the approximative character of the truncated theory we assume, as usual,the relationζ=ρτthat guarantees aﬁnite propagation velocity of viscous pulses[10,27-29,31].In the following subsection we are going to generalize this ing(7)and(19)in equation(11)withζ=ρτwe ﬁndπ+τ˙π=−ρΘτ 1+π2ρ2(a+1) ,(20) with the abbreviationsa≡ρn∂ρ∂ρ/∂T=1∂T,γ=ρ+p2ρ(1+b+γ)≪1,π23T≫1however,the second condition has to be checked separately.The inequalities (22)may be regarded as criteria under which the truncated theory is a reasonable approximation to the full theory.Given equations of state(15) and(16),any solutionπof the truncated theory may be tested according to (22)whether or not and to which accuracy it approximates the full theory.But there is a diﬀerent possibility,namely the caseπ2ρ2(a+1)=0,(23)in which all the terms that distinguish the full from the truncated theory cancel among themselves.Relation(23)is identical to(14)forζ=ρτ,i.e., it is the condition under which the full theory is identical to the truncated one.Solving(23)forπ/ρyieldsπ1+a.(24)7According to the above mentioned parameter ranges forγ,b and a,|π|≪ρis only possible for a≫1,i.e.,for massive ing(24)in(7)yields˙ρ1+a.(25)For radiation(γ=4/3,b=1/3,a=1)weﬁnd˙ρ=0and,consequently,Θ=Θ0=const in aﬂat FLRW universe.The truncated and the full theory coincide in a speciﬁc bulk viscosity driven inﬂationary universe.Since with (24)πis completely determined byρ,provided,the equations of state are given,the remaining equation(13)is no longer a dynamical equation on its own,but may be used to calculateτ=τ(Θ0).Since the solution is stationary,i.e.,˙π=0,weﬁndτr0=4T =−Θb−a(γ+1)T=Θ(28)for radiation,or T∼R3.The temperature increases in an expanding uni-verse.This implication of the condition(14)wasﬁrst noticed by Maartens [29].While this might appear strange at theﬁrst glance,it is an unavoidable feature of any bulk viscosity driven inﬂation as is obvious from(19).This point will be discussed in more detail in the following subsection.2.3The dynamics of the full second order theoryIn this subsection we shall investigate the full causal theory assuming the existence of general equations of state(15),(16)andζFollowing Belinskii et al.[10]usually the relationζ/τ=ρwas used to guar-antee that the propagation velocity of viscous pulses,which is expected to be of the order[28]v∼ ζf =−Θρf′ρ .(31)Together with(19),equation(11)may now be written asπ+τ˙π=−ρΘτ f2ρ 1+b+γρf′2ρ2 a+ρf′3=κρ,(33) whereκis Einstein’s gravitational constant,and˙Θ=−3κρ=−γ−2H2.(36)From(15),using(6)and(19)one realizes˙p=v2s˙ρ+Θπ v2s−b ,(37)9with the sound velocity v s ,given byv 2s = ∂pρ+p ∂p ρ+p (∂p/∂T )2Hτ a +ρ2ρ2(1+b )+˙H −9ρ+γ2γH 2=0.(40)It should be pointed out again that in arriving at this evolution equation only equations of state (15),(16)and (29)were used.We think this set of equations of state to be more general than those used in previous papers[31,28,29].Especially,no speciﬁc temperature law like T =βρr in [31]had to be postulated.There is no freedom to impose a separate temperature law.The behaviour of the temperature is generally governed by (19).As it is obvious from (18),a relation T ∼ρr is possible for ∂ρ/∂n =0.Even in this case,however,r is not arbitrary but determined by r =ρ(T dρ/dT )−1.In the speciﬁc case of a ﬂat FRLW universe with˙H2˙ρT =3(γa −b )˙Rρ.(42)For constant values of a ,b and γT ∼ρa R 3(γa −b )(43)results,which in the radiation dominated case with a =1,b =1/3,γ=4/3,speciﬁes toT r ∼ρr R 3.(44)Obviously,with ρr ∼R −4for π=0,one obtains the correct limiting case T r ∼R −1.102.4Viscous exponential inﬂationIt has been a matter of some debate whether a bulk viscous pressure is able to drive inﬂation[26-29,31].While Hiscock and Salmonson[26]have shown that,using the equations of state for a Boltzmann gas,the full causal theory does not admit an inﬂationary phase,diﬀerent equations of state like those used by Romano and Pav´o n[31],Zakari and Jou[28]and Maartens[29]are compatible with a de Sitter phase.It should be stressed again that dealing with the question of a viscosity driven inﬂation one has to assume that the MIS theory is applicable far from equilibrium[29].Under these premises we are going to consider now the possibility of a solution of the evolution equation for H with H=H0=const.In such a case eq.(40)yieldsτ−10=3H0 f2(γa−1−b)+v2s−b .(45) In the radiation dominated case this reduces toτr0=4fH−10.(46)Keeping in mind that until now only f=f(ρ)was used,one recognizes that the ratio betweenτr0and H−10crucially depends on f/ρ.If,as is usually assumed,f=ρ,we haveτr0<H−10.If,however,f is allowed to be smaller thanρ,we may haveτr0>H−10.It follows from(30)that a relation f<ρimplies a lower propagation velocity compared with the case f=ρ.Provided f=ρis equivalent to a propagation velocity that coincides with the velocity of light,f=ρ/3,e.g.,leads to a propagation with v s=1/√3H−10,(47) implyingτr0>H−10.In this case the nonequilibrium is‘frozen in’.The viscous inﬂation is‘nonthermalizing’[29].While forτr0<H−10there is a quick(compared with the expansion rate)relaxation to(local)equilibrium, this is no longer the case ifτr0becomes comparable to the expansion rate H−10.Nonequilibrium situations like this may occur,e.g.,in GUTs close to the Planck time where the underlying microscopic process is the decay of heavy vector bosons[32].According to Maartens[29]the conditionτ0> H−10may be regarded as a consistency criterion for causal viscous inﬂation. Clearly,forτ0<H−10,the Universe will relax to an equilibrium state in less11than one expansion time.A successful inﬂation,however,has to last for many characteristic expansion times.In the matter dominated case withγ=1,b=2/3,v2s≪1,a=2mH−10,(48)mi.e.,we are always in the rangeτm0≪H−10.The time scale for the relaxation to equilibrium is much smaller than the expansion time scale.Calculating the dependenceτ0(H0)along the the same lines that lead to(45)within the truncated version provides us withτ−10=3γ−1 f=−γ.(50)ρAccording to(6)the particle number density decreases exponentially.The temperature evolution law(19)together with(50)becomes˙Tρ∂ρ/∂T−∂ρc vinﬂationary phase(see below),this point deserves a detailed discussion.It is frequently used that in the equations of state for radiation in equilibrium, p=nT andρ=3nT,which are speciﬁc cases of(15)and(16),the number density n may be eliminated according to n∼T3,resulting in p=p(T) andρ=ρ(T)with p=ρ/3andρ∼T4.Then T is the only independent variable and∂ρ/∂T→dρ/dT.In order to check whether a corresponding procedure is possible forπ=0,we shall assume an arbitrary dependence n=n(T)instead of the equilibrium relation n∼T3.With(19)weﬁnd˙n=−3HT dnρ .(53)Comparison with(6)leads to the general relationdTρ dnstate(16).Since n decreases exponentially,T must increase accordingly in order to guarantee˙ρ=0,as long as∂ρ/∂n>0and∂ρ/∂T>0.Exactly this kind of behaviour was found by Hiscock and Salmonson[26]for the trun-cated Israel-Stewart theory.As was demonstrated above,the full theory exhibits a corresponding feature as well.(Note,that according to[26]there does not exist a de Sitter phase in the full theory for the case of a Boltzmann gas).If one does not use,however,a relationship T∼ρr and the behaviour of T is governed by(52)with∂ρ/∂n>0,this has important consequences for the entropy production during bulk viscous driven inﬂation,discussed by Maartens[29].The entropy in a comoving volume isΣ=nsR3.With (6)and(9)the change ofΣis˙Σ=−πΘ.(58)TThis change of the comoving entropy depends crucially on the behaviour of T.For T=const which follows from T∼ρr,equivalent to∂ρ/∂n=0(see (52)),we recover the exponential increase of˙Σfound by Maartens[29].For equations of state(15)and(16)however,with∂ρ/∂n>0,the temperature dependence in the case of radiation(γ=4/3)is T r∼ρ0R3as follows from(44).˙Σturns out to be constant rather than exponentially increasing. Integrating(58)oneﬁnds only a linear growth inΣ.Consequently,there is no way to generate a considerable amount of entropy during a bulk viscosity driven de Sitter phase.To investigate the stability of the solutions H=H0,we probe the latter with small perturbations,i.e.,we assume H=H0(1+h(t))with|h|≪1in eq.(40).Equation(40)is valid for spatiallyﬂat homogeneous and isotropic spacetimes,therefore the stability analysis is restricted to this case as well. We assume that the dimensionless quantities f/ρ,γ,a,b and v s remain unchanged for small deviations from H=H0.Since we found the relaxation timeτ0in the inﬂationary phase to be proportional to H−10it is natural to assumeτ∼H−1generally and toﬁx the proportionality factor by(45). Retaining only terms linear in h,the resulting equation for h is¨h+3H˙h=0,(59)0K14withK≡1+f2 ρnot change the overall number ofﬂuid particles,i.e.,it is assumed that(4) holds.However,interactions with conserved particle numbers are only a spe-cial case,particularly at high energies.Moreover,at times of the order of the Planck time or during the scalarﬁeld decay in inﬂationary scenarios,par-ticle or string production processes are supposed to aﬀect the cosmological dynamics,leading to features like‘deﬂationary universes’and‘string-driven inﬂation’[16-20,38].In this section we generalize the previous formalism to the case that the existence of a nonvanishing bulk pressure is accompanied by an increase in the number ofﬂuid particles.We start our investigations with(1),(2),(3) and(5)as well,but(4)is now replaced byN a;a=nΓ,(61)yielding˙n+Θn=nΓ.(62)Γis the particle production rate which has to be regarded as an input quantity in our phenomenological description.Instead of(9)nownT˙s=−Θπ−(ρ+p)Γ(63)results.The entropy production density is given byT S a;a=−nµΓ−π Θ+τ2πT τ.(65)nA change in the number of particles,i.e.,Γ=0,is believed to be phe-nomenologically equivalent to an eﬀective viscous pressure[13,15,17-21].A discussion of this equivalence on the level of relativistic kinetic theory has been given recently[33].In order to relate the present investigations to previous work we shall in aﬁrst step focus on the Eckart theory,i.e.,to the caseτ=0.Equation(64)reduces toT S a E;a=−nµΓ−πEΘ(66)in this case,where the subscript E stands for‘Eckart’.Formal rewriting yieldsT S a E;a=−πE Θ+nµπ−1EΓ .(67)16As usual,one has to guarantee S a E ;a ≥0.Since we expect the particle production to be eﬀectively equivalent to an viscous pressure we have todemand additionally that the entire entropy production is given in terms of πE ,i.e.,S a E ;a =π2Eρ+p .(74)17Combining the latter relation again with(72)weﬁnd that the partλζof the generalized coeﬃcient of bulk viscosityζthat arises due to a nonvanishing particle production rate isλζ=ΓΘ−2ζ˙π+1ζTu a ;a+µnΓ2π2T τwhich follows from the requirement that the creation process does not aﬀect the entropy per particle has to be abandoned,this means that within the causal theory the creation process contributes to˙s in general.While in the Eckart theory the particle production rate was rather simply related to the viscous pressure by(72),Γenters the equation forπof the causal theory as an independent parameter in a less obvious way.Our present approach assumes that the deviations from equilibrium may be characterized in terms of one single quantityπalso for the case of a non-vanishing particle production rateΓ.This implies that there is only one (generalized)coeﬃcient of bulk viscosityζand one relaxation timeτ.The advantage of this assumption is that there exists only one causal evolution equation,namely(77),taking into account the inﬂuence ofΓonπ.(For an alternative proposal see again[31]).Since the latter equation is nonlin-ear there is no obvious separation into a conventional bulk pressure and a ‘creation’pressure.We shall assume furtheron,thatζandτcontinue to be related by(29).The dependence of the relaxation timeτonΓwill be discussed below for speciﬁc cases.Forµ=0eq.(77)appears to be identical to the caseΓ=0,but there are diﬀerences in the behaviour of the particle number density and the tem-perature.The former is determined by(62).Using(16),(17),(7)and(62) instead of(6),the temperature law(19)is replaced by(cf.[22])˙T∂ρ/∂T +π∂ρ/∂T−ρ+pT =−(Θ−Γ)∂p/∂TT∂ρ/∂T=−(Θ−Γ)∂p/∂T∂ρ/∂T.(79)With this evolution law for the temperature,diﬀerent from(19)the evolution equation(77)forπbecomes diﬀerent from(32)even in the caseµ=0, although(10)and(76)seem to coincide.The relation(37)for˙p is now replaced by˙p=ρ (Γ−Θ)γv2s−b γΓ+ΘπInstead of (39)we haveκ˙π=−2¨H −6H ˙H (1+b )+9H 3γ v 2s −b −3H 2Γγ v 2s −b.(81)The evolution equation for πmay be written π+τ˙π=−ρτ f ρ+pγ2ρ Θ 1+b +γρf ′2ρ2Θ a +ρf ′H τ a +ρ2ρ2(1+b )+˙H +12H 3τ f2(γa −1−b )+γ v 2s −b+32 +3γH 23˙H 2H 2Γτfρ+p =0.(83)For an easier comparison with eq.(40)we kept separate the common factor π/ρ(see (36))on the l.h.s.of this equation.Having only one evolution equation for H is a consequence of our previous assumption that it is possible to characterize the deviations from equilibrium by a single quantity πonly.Eq.(83)simpliﬁes considerably for µ=0:τ¨H −˙H 2f f ′ −3˙HHτ γa +γf f ′−32Γ˙Hτ(γa −b )−9ρ+γ+32 +3T=3(γa−b)˙Rρ.(85)instead of(42).For constant values of a,b andγT∼ρa R3n γa−b.(86) Forγ=4/3,b=1/3,a=1,this speciﬁes toT∼ρR3n.(87)3.2Inﬂationary solutionsLooking for solutions H=H0=const of(83),weﬁnd the following expres-sion forτ0that generalizes(45)τ−10=3H0 f2+v2s−b−Γ nµγρ+γa−bγρ 1−λnµ2+(1−λ)γa−bFor particles withµ=0and an equation of state close to that for radiation it is obvious that for anyλ>0the relaxation timeτ0is larger than forλ=0. In the limiting caseλ=1the production rate of the particles coincides with the expansion rate.τ0then is given byτ−10(λ=1)=3H0 fρ+p −19H−10ρ−1ornT ˙s =ρR 3γR 3γ.(93)With (62)andT =T 0 ρR 30N γa −b(94)from (86),where the subscript 0refers to some initial time,one ﬁnds˙s =1ρ aR 30N R 3 γ ρR 3γn 0T 0γ R 30N n 0T 0H 0−1n 0T 0 H 0(t −t 0)−1n 0T 0 H 0(t −t 0)−1n 0T 0(1−λ)H 0(t −t 0)+s (t 0) N (t 0)exp [3λH 0(t −t 0)].(100)It follows that for Γ=0we have an exponential increase of the comoving entropy Σduring the de Sitter phase.Starting from the latter expression for23Σit is possible to apply Maartens’numerical estimations[29]concerning the entropy production during the inﬂationary phase.All his considerations of this point are valid in the present case forλ=0,provided his H0is replaced byλH0.His conclusion that it is possible‘to generate the right amount of entropy without re-heating’may hold in a universe with particle production. There is complete equivalence to Maartens’result for the entropy production forλ=1.Consequently,a substantial production of entropy during a(bulk viscosity induced)dissipational inﬂationary phase is possible if at least a part of the bulk viscous pressure is related to an increase in the particle number.In a sense we have reestablished Maartens’result,although within a diﬀerent setting:An exponential growth of the comoving entropy is only possible if the viscous pressureπin(1)is,at least partially,a‘creation’pressure connected with an increase in the number ofﬂuid particles rather than an increase in the entropy per particle.It remains to consider the stability of the inﬂationary solutions forΓ=0. Proceeding as in section2we have¨h+(3H0K−ΓM)˙h=0(101) instead of(59),whereM≡2fρ+p+v2s−b.(102)Consequently,the inﬂationary solutions are stable for3H0K−ΓM>0. Forµ=0and the equations of state for radiation one has M=0and the stability condition reduces to K>0again.The stability properties of the corresponding inﬂationary solutions are not aﬀected at all by a nonvanishing Γ.With equations of state for matter the inﬂationary solutions have the same instability as forΓ=0.4SummaryUsing particle number density and temperature as basic thermodynamical variables of the cosmicﬂuid we have studied the full M¨u ller-Israel-Stewart theory for a spatiallyﬂat,homogeneous and isotropic universe with bulk viscous pressure.We found general criteria for the applicability of the so called truncated versions.It was shown that in exceptional cases there exist common solutions of the full and the truncated theories far from equilibrium. The possibility of exponential inﬂationary solutions driven by bulk viscosity24was investigated.Almost all corresponding solutions imply an exponential growth of the temperature during the de Sitter stage.Since the number density of theﬂuid particles decreases exponentially,only a corresponding increase in the temperature guarantees a constant energy density as long as the speciﬁc heat is positive andﬁnite and the energy density increases with the particle number density.In the second part of the paper the bulk viscous pressure was allowed to account partially or fully for particle pro-duction processes.A nonvanishing particle production rate may enlarge the relaxation time for a viscous pressure that is supposed to drive inﬂation.It may help to‘freeze in’the corresponding nonequilibrium,i.e.,to improve the conditions for inﬂation.There exist stable,inﬂationary solutions for equa-tions of state close to that for radiation.Due to the increase in the particle number the comoving entropy increases exponentially in this period.Only in the limiting case that the entire inﬂation driving viscous pressure is due to particle production both the number density and the temperature remain constant during the de Sitter phase.AcknowledgmentThis paper was inspired by a preprint of[29].I thank Roy Maartens, Portsmouth,for clarifying conversations.I am indebted to the members of the Grup de F´ısica Estad´ıstica of the Autonomous University of Barcelona for their warm hospitality.Helpful discussions with Diego Pav´o n who crit-ically commented on previous versions of this paper,David Jou and Josep Triginer are gratefully acknowledged.References[1]C.Eckart,Phys.Rev.58,919(1940).[2]ndau and E.M.Lifshitz,Fluid Mechanics,(Addison-Wesley,Reading,1958).[3]I.M¨u ller,Z.Physik198,329(1967).[4]W.Israel,Ann.Phys.(NY)100,310(1976).[5]W.Israel and J.M.Stewart,Ann.Phys.(NY)118,341(1979).[6]W.Israel and J.M.Stewart,Proc.R.Soc.Lond.A365,43(1979).[7]D.Pav´o n,D.Jou and J.Casas-V´a zquez,Ann.Inst.H.Poincar´e Sec.A36,79(1982).[8]W.A.Hiscock and L.Lindblom,Ann.Phys.(NY)151,466(1983).[9]W.A.Hiscock and L.Lindblom,Phys.Rev.D35,3723(1987).25。

Nambu-Goto string with massive ends at finite temperature

PACS numbers: 12.38.Aw, 12.40.-y, 12.39.Pn Key words: relativistic string, quarks at the string ends, critical temperature, interquark potential, quark masses.
R→∞
The string potential at ﬁnite temperature is calculated directly in the same way as at T = 0 [1]. As a result one obtains the well known square-root expression
2
where σ (T ) is an eﬀective string tension depending on the temperature T . At T = Tc the string tension vanishes σ (Tc ) = lim R−1 V (R, Tc ) = 0 . (2.2)
2
Critical temperature in string models
The critical temperature (or the temperature of deconﬁnement) in string models is determined in the following way. If V (R, T ) is the string potential calculated at ﬁnite temperature T then its asymptotics at large distances is V (R, T ) → σ (T ) · R,

综述BEC(Bose-Einstein Condensation) 【英文版】

Theory of Bose-Einstein condensation in trapped gasesFranco Dalfovo and Stefano GiorginiDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,ItalyLev P.PitaevskiiDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,Italy;Department of Physics,TECHNION,Haifa32000,Israel;and Kapitza Institute for Physical Problems,ul.Kosygina2,117334MoscowSandro StringariDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,ItalyThe phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective.Mean-ﬁeld theory provides a framework to understand the main features of the condensation and the role of interactions between particles.Various properties of these systems are discussed,including the density proﬁles and the energy of the ground-state conﬁgurations,the collective oscillations and the dynamics of the expansion,the condensate fraction and the thermodynamic functions.The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales.Despite the dilute nature of the gases,interactions profoundly modify the static as well as the dynamic properties of the system;the predictions of mean-ﬁeld theory are in excellent agreement with available experimental results.Effects of superﬂuidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed,as well as the consequences of coherence such as the Josephson effect and interference phenomena.The review also assesses the accuracy and limitations of the mean-ﬁeld approach.[S0034-6861(99)00103-8]CONTENTSI.Introduction463II.The Ideal Bose Gas in a Harmonic Trap466A.The condensate of noninteracting bosons466B.Trapped bosons atﬁnite temperature:thermodynamic limit468C.Finite-size effects470D.Role of dimensionality471E.Nonharmonic traps and adiabatic transformations472 III.Effects of Interactions:Ground State472A.Order parameter and mean-ﬁeld theory472B.Ground state474C.Collapse for attractive forces477rge-N limit for repulsive forces478E.Beyond mean-ﬁeld theory479 IV.Effects of Interactions:Dynamics480A.Excitations of the condensate and time-dependent Gross-Pitaevskii equation480rge Na/a ho limit and collisionlesshydrodynamics482C.Sum rules and collective excitations485D.Expansion and large-amplitude oscillations486E.Density of states:collective vs single-particleexcitations489 V.Effects of Interactions:Thermodynamics491A.Relevant energy scales491B.Critical temperature492C.Below T c493D.Thermodynamic limit and scaling495E.Results for the thermodynamic functions496F.Collective modes atﬁnite temperature498 VI.Superﬂuidity and Coherence Phenomena500A.Rotational properties:vortices and moment ofinertia501B.Interference and Josephson effect504C.Collapse and revival of collective oscillations506 VII.Conclusions and Outlook507 Acknowledgments509 References509 I.INTRODUCTIONBose-Einstein condensation(BEC)(Bose,1924;Ein-stein,1924,1925)was observed in1995in a remarkable series of experiments on vapors of rubidium(Anderson et al.,1995)and sodium(Davis et al.,1995)in which the atoms were conﬁned in magnetic traps and cooled down to extremely low temperatures,of the order of fractions of microkelvins.Theﬁrst evidence for condensation emerged from time-of-ﬂight measurements.The atoms were left to expand by switching off the conﬁning trap and then imaged with optical methods.A sharp peak in the velocity distribution was then observed below a cer-tain critical temperature,providing a clear signature for BEC.In Fig.1,we show one of theﬁrst pictures of the atomic clouds of rubidium.In the same year,ﬁrst signa-tures of the occurrence of BEC in vapors of lithium were also reported(Bradley et al.,1995).Though the experiments of1995on the alkalis should be considered a milestone in the history of BEC,the experimental and theoretical research on this unique phenomenon predicted by quantum statistical mechanics is much older and has involved different areas of physics (for an interdisciplinary review of BEC see Grifﬁn, Snoke,and Stringari,1995).In particular,from the very beginning,superﬂuidity in helium was considered by463Reviews of Modern Physics,Vol.71,No.3,April19990034-6861/99/71(3)/463(50)/$25.00©1999The American Physical SocietyLondon (1938)as a possible manifestation of BEC.Evi-dence for BEC in helium later emerged from the analy-sis of the momentum distribution of the atoms measured in neutron-scattering experiments (Sokol,1995).In re-cent years,BEC has been also investigated in the gas of paraexcitons in semiconductors (see Wolfe,Lin,and Snoke,1995,and references therein),but an unambigu-ous signature for BEC in this system has proven difﬁcult to ﬁnd.Efforts to Bose condense atomic gases began with hy-drogen more than 15years ago.In a series of experi-ments hydrogen atoms were ﬁrst cooled in adilutionFIG.2.(Color)Collective excitations of a Bose-Einstein condensate.Shown are in situ repeated phase-contrast images taken of a ‘‘pure’’condensate.The excitations were produced by modulating the magnetic ﬁelds which conﬁne the condensate,and then letting the condensate evolve freely.Both the center-of-mass and the shape oscillations are visible,and the ratio of their oscillation frequencies can be accurately measured.The ﬁeld of view in the vertical direction is about 620␮m,corresponding to a condensate width of the order of 200–300␮m.The time step is 5ms per frame.From Stamper-Kurn and Ketterle(1998).FIG.1.(Color)Images of the velocity distribution of rubidium atoms in the experiment by Anderson et al.(1995),taken by means of the expansion method.The left frame corresponds to a gas at a temperature just above condensation;the center frame,just after the appearance of the condensate;the right frame,after further evaporation leaves a sample of nearly pure condensate.The ﬁeld of view is 200␮m ϫ270␮m,and corresponds to the distance the atoms have moved in about 1/20s.The color corresponds to the number of atoms at each velocity,with red being the fewest and white being the most.From Cornell (1996).464Dalfovo et al.:Bose-Einstein condensation in trapped gasesRev.Mod.Phys.,Vol.71,No.3,April 1999refrigerator,then trapped by a magneticﬁeld and fur-ther cooled by evaporation.This approach has come very close to observing BEC,but is still limited by re-combination of individual atoms to form molecules(Sil-vera and Walraven,1980and1986;Greytak and Klepp-ner,1984;Greytak,1995;Silvera,1995).At the time of this review,ﬁrst observations of BEC in spin-polarized hydrogen have been reported(Fried et al.,1998).In the 1980s laser-based techniques,such as laser cooling and magneto-optical trapping,were developed to cool and trap neutral atoms[for recent reviews,see Chu(1998), Cohen-Tannoudji(1998),and Phillips(1998)].Alkali at-oms are well suited to laser-based methods because their optical transitions can be excited by available lasers and because they have a favorable internal energy-level structure for cooling to very low temperatures.Once they are trapped,their temperature can be lowered fur-ther by evaporative cooling[this technique has been re-cently reviewed by Ketterle and van Druten(1996a)and by Walraven(1996)].By combining laser and evapora-tive cooling for alkali atoms,experimentalists eventually succeeded in reaching the temperatures and densities re-quired to observe BEC.It is worth noticing that,in these conditions,the equilibrium conﬁguration of the system would be the solid phase.Thus,in order to observe BEC,one has to preserve the system in a metastable gas phase for a sufﬁciently long time.This is possible be-cause three-body collisions are rare events in dilute and cold gases,whose lifetime is hence long enough to carryout experiments.So far BEC has been realized in87Rb(Anderson et al.,1995;Han et al.,1998;Kasevich,1997;Ernst,Marte et al.,1998;Esslinger et al.,1998;So¨dinget al.,1999),in23Na(Davis et al.,1995;Hau,1997and 1998;Lutwak et al.,1998),and in7Li(Bradley et al.,1995and1997).The number of experiments on BEC invapors of rubidium and sodium is now growing fast.Inthe meanwhile,intense experimental research is cur-rently carried out also on vapors of caesium,potassium,and metastable helium.One of the most relevant features of these trappedBose gases is that they are inhomogeneous andﬁnite-sized systems,the number of atoms ranging typicallyfrom a few thousands to several millions.In most cases,the conﬁning traps are well approximated by harmonicpotentials.The trapping frequency␻ho also provides a characteristic length scale for the system,a ho ϭ͓ប/(m␻ho)͔1/2,of the order of a few microns in the available samples.Density variations occur on this scale.This is a major difference with respect to other systems,like,for instance,superﬂuid helium,where the effects ofinhomogeneity take place on a microscopic scaleﬁxedby the interatomic distance.In the case of87Rb and 23Na,the size of the system is enlarged as an effect of repulsive two-body forces and the trapped gases can be-come almost macroscopic objects,directly measurable with optical methods.As an example,we show in Fig.2 a sequence of in situ images of an oscillating condensate of sodium atoms taken at the Massachusetts Institute of Technology(MIT),where the mean axial extent is of the order of0.3mm.The fact that these gases are highly inhomogeneous has several important consequences.First BEC shows up not only in momentum space,as happens in super-ﬂuid helium,but also in coordinate space.This double possibility of investigating the effects of condensation is very interesting from both the theoretical and experi-mental viewpoints and provides novel methods of inves-tigation for relevant quantities,like the temperature de-pendence of the condensate,energy and density distributions,interference phenomena,frequencies of collective excitations,and so on.Another important consequence of the inhomogene-ity of these systems is the role played by two-body inter-actions.This aspect will be extensively discussed in the present review.The main point is that,despite the very dilute nature of these gases(typically the average dis-tance between atoms is more than ten times the range of interatomic forces),the combination of BEC and har-monic trapping greatly enhances the effects of the atom-atom interactions on important measurable quantities. For instance,the central density of the interacting gas at very low temperature can be easily one or two orders of magnitude smaller than the density predicted for an ideal gas in the same trap,as shown in Fig.3.Despite the inhomogeneity of these systems,which makes the solution of the many-body problem nontrivial,the dilute nature of the gas allows one to describe the effects of the interaction in a rather fundamental way.In practice a single physical parameter,the s-wave scattering length, is sufﬁcient to obtain an accuratedescription.FIG.3.Density distribution of80000sodium atoms in the trap of Hau et al.(1998)as a function of the axial coordinate.The experimental points correspond to the measured optical den-sity,which is proportional to the column density of the atom cloud along the path of the light beam.The data agree well with the prediction of mean-ﬁeld theory for interacting atoms (solid line)discussed in Sec.III.Conversely,a noninteracting gas in the same trap would have a much sharper Gaussian distribution(dashed line).The same normalization is used for the three density proﬁles.The central peak of the Gaussian is found at about5500␮mϪ2.Theﬁgure points out the role of atom-atom interaction in reducing the central density and en-larging the size of the cloud.465Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999The recent experimental achievements of BEC in al-kali vapors have renewed a great interest in the theoret-ical studies of Bose gases.A rather massive amount of work has been done in the last couple of years,both to interpret the initial observations and to predict new phe-nomena.In the presence of harmonic conﬁnement,the many-body theory of interacting Bose gases gives rise to several unexpected features.This opens new theoretical perspectives in this interdisciplinaryﬁeld,where useful concepts coming from different areas of physics(atomic physics,quantum optics,statistical mechanics,and condensed-matter physics)are now merging together. The natural starting point for studying the behavior of these systems is the theory of weakly interacting bosons which,for inhomogeneous systems,takes the form of the Gross-Pitaevskii theory.This is a mean-ﬁeld ap-proach for the order parameter associated with the con-densate.It provides closed and relatively simple equa-tions for describing the relevant phenomena associated with BEC.In particular,it reproduces typical properties exhibited by superﬂuid systems,like the propagation of collective excitations and the interference effects origi-nating from the phase of the order parameter.The theory is well suited to describing most of the effects of two-body interactions in these dilute gases at zero tem-perature and can be naturally generalized to also ex-plore thermal effects.An extensive discussion of the application of mean-ﬁeld theory to these systems is the main basis of the present review article.We also give,whenever possible, simple arguments based on scales of length,energy,and density,in order to point out the relevant parameters for the description of the various phenomena.There are several topics which are only marginally discussed in our paper.These include,among others, collisional and thermalization processes,phase diffusion phenomena,light scattering from the condensate,and analogies with systems of coherent photons.In this sense our work is complementary to other recent review ar-ticles(Burnett,1996;Parkins and Walls,1998).Further-more,in our paper we do not discuss the physics of ul-tracold collisions and the determination of the scattering length which have been recently the object of important experimental and theoretical studies in the alkalis (Heinzen,1997;Weiner et al.,1999).The plan of the paper is the following:In Sec.II we summarize the basic features of the non-interacting Bose gas in harmonic traps and we introduce theﬁrst relevant length and energy scales,like the oscil-lator length and the critical temperature.We also com-ment onﬁnite-size effects,on the role of dimensionality and on the possible relevance of anharmonic traps.In Sec.III we discuss the effects of the interaction on the ground state.We develop the formalism of mean-ﬁeld theory,based on the Gross-Pitaevskii equation.We consider the case of gases interacting with both repulsive and attractive forces.We then discuss in detail the large-N limit for systems interacting with repulsive forces,leading to the so-called Thomas-Fermi approxi-mation,where the ground-state properties can be calcu-lated in analytic form.In the last part,we discuss the validity of the mean-ﬁeld approach and give explicit re-sults for theﬁrst corrections,beyond meanﬁeld,to the ground-state properties,including the quantum deple-tion of the condensate,i.e.,the decrease in the conden-sate fraction produced by the interaction.In Sec.IV we investigate the dynamic behavior of the condensate using the time-dependent Gross-Pitaevskii equation.The equations of motion for the density and the velocityﬁeld of the condensate in the large-N limit, where the Thomas-Fermi approximation is valid,are shown to have the form of the hydrodynamic equations of superﬂuids.We also discuss the dynamic behavior in the nonlinear regime(large amplitude oscillations and free expansion),the collective modes in the case of at-tractive forces,and the transition from collective to single-particle states in the spectrum of excitations.In Sec.V we discuss thermal effects.We show how one can deﬁne the thermodynamic limit in these inho-mogeneous systems and how interactions modify the be-havior compared to the noninteracting case.We exten-sively discuss the occurrence of scaling properties in the thermodynamic limit.We review several results for the shift of the critical temperature and for the temperature dependence of thermodynamic functions,like the con-densate fraction,the chemical potential,and the release energy.We also discuss the behavior of the excitations atﬁnite temperature.In Sec.VI we illustrate some features of these trapped Bose gases in connection with superﬂuidity and phase coherence.We discuss,in particular,the structure of quantized vortices and the behavior of the moment of inertia,as well as interference phenomena and quantum effects beyond mean-ﬁeld theory,like the collapse re-vival of collective oscillations.In Sec.VII we draw our conclusions and we discuss some further future perspectives in theﬁeld.The overlap between current theoretical and experi-mental investigations of BEC in trapped alkalis is al-ready wide and rich.Various theoretical predictions, concerning the ground state,dynamics,and thermody-namics are found to agree very well with observations; others are stimulating new experiments.The comparison between theory and experiments then represents an ex-citing feature of these novel systems,which will be fre-quently emphasized in the present review.II.THE IDEAL BOSE GAS IN A HARMONIC TRAPA.The condensate of noninteracting bosonsAn important feature characterizing the available magnetic traps for alkali atoms is that the conﬁning po-tential can be safely approximated with the quadratic formV ext͑r͒ϭm2͑␻x2x2ϩ␻y2y2ϩ␻z2z2͒.(1) Thus the investigation of these systems starts as a text-466Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999book application of nonrelativistic quantum mechanics for identical pointlike particles in a harmonic potential. Theﬁrst step consists in neglecting the atom-atom in-teraction.In this case,almost all predictions are analyti-cal and relatively simple.The many-body Hamiltonian is the sum of single-particle Hamiltonians whose eigenval-ues have the form␧nx n y n zϭͩn xϩ12ͪប␻xϩͩn yϩ12ͪប␻yϩͩn zϩ12ͪប␻z,(2)where͕n x,n y,n z͖are non-negative integers.The ground state␾(r1,...,r N)of N noninteracting bosons conﬁned by the potential(1)is obtained by putting all the particles in the lowest single-particle state(n xϭn y ϭn zϭ0),namely␾(r1,...,r N)ϭ⌸i␸0(r i),where␸0(r) is given by␸0͑r͒ϭͩm␻ho␲បͪ3/4expͫϪm2ប͑␻x x2ϩ␻y y2ϩ␻z z2͒ͬ,(3)and we have introduced the geometric average of the oscillator frequencies:␻hoϭ͑␻x␻y␻z͒1/3.(4) The density distribution then becomes n(r)ϭN͉␸0(r)͉2 and its value grows with N.The size of the cloud is in-stead independent of N and isﬁxed by the harmonic oscillator length:a hoϭͩបm hoͪ1/2(5)which corresponds to the average width of the Gaussian in Eq.(3).This is theﬁrst important length scale of the system.In the available experiments,it is typically of the order of a hoϷ1␮m.Atﬁnite temperature only part of the atoms occupy the lowest state,the others being ther-mally distributed in the excited states at higher energy. The radius of the thermal cloud is larger than a ho.A rough estimate can be obtained by assuming k B T ӷប␻ho and approximating the density of the thermal cloud with a classical Boltzmann distribution n cl(r)ϰexp͓ϪV ext(r)/k B T͔.If V ext(r)ϭ(1/2)m␻ho2r2,the width of the Gaussian is R Tϭa ho(k B T/ប␻ho)1/2,and hence larger than a ho.The use of a Bose distribution function does not change signiﬁcantly this estimate. The above discussion reveals that Bose-Einstein con-densation in harmonic traps shows up with the appear-ance of a sharp peak in the central region of the density distribution.An example is shown in Fig.4,where we plot the prediction for the condensate and thermal den-sities of5000noninteracting particles in a spherical trap at a temperature Tϭ0.9T c0,where T c0is the temperature at which condensation occurs(see discussion in the next section).The curves correspond to the column density, namely the particle density integrated along one direc-tion,n(z)ϭ͐dx n(x,0,z);this is a typical measured quantity,the x direction being the direction of the light beam used to image the atomic cloud.By plotting di-rectly the density n(r),the ratio of the condensed and noncondensed densities at the center would be even larger.By taking the Fourier transform of the ground-state wave function,one can also calculate the momentum distribution of the atoms in the condensate.For the ideal gas,it is given by a Gaussian centered at zero mo-mentum and having a width proportional to a hoϪ1.The distribution of the thermal cloud is,also in momentum space,ing a classical distribution function oneﬁnds that the width is proportional to(k B T)1/2.Ac-tually,the momentum distributions of the condensed and noncondensed particles of an ideal gas in harmonic traps have exactly the same form as the density distribu-tions n0and n T shown in Fig.4.The appearance of the condensate as a narrow peak in both coordinate and momentum space is a peculiar fea-ture of trapped Bose gases having important conse-quences in both the experimental and theoretical analy-sis.This is different from the case of a uniform gas where the particles condense into a state of zero mo-mentum,but BEC cannot be revealed in coordinate space,since the condensed and noncondensed particles ﬁll the same volume.Indeed,the condensate has been detected experimen-tally as the occurrence of a sharp peak over a broader distribution,in both the velocity and spatial distribu-tions.In theﬁrst case,one lets the condensate expand freely,by switching off the trap,and measures the den-sity of the expanded cloud with light absorption(Ander-son et al.,1995).If the particles do not interact,the ex-pansion is ballistic and the imaged spatial distribution of the expanding cloud can be directly related to the initial momentum distribution.In the second case,one mea-sures directly the density of the atoms in the trapby FIG.4.Column density for5000noninteracting bosons in a spherical trap at temperature Tϭ0.9T c0.The central peak is the condensate,superimposed on the broader thermal distri-bution.Distance and density are in units of a ho and a hoϪ2,re-spectively.The density is normalized to the number of atoms. The same curves can be identiﬁed with the momentum distri-bution of the condensed and noncondensed particles,provided the abscissa and the ordinate are replaced with p z,in units of a hoϪ1,and the momentum distribution,in units of a ho2,respec-tively.467Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999means of dispersive light scattering(Andrews et al., 1996).In both cases,the appearance of a sharp peak is the main signature of Bose-Einstein condensation.An important theoretical task consists of predicting how the shape of these peaks is modiﬁed by the inclusion of two-body interactions.As anticipated in Fig.3,the interac-tions can change the picture drastically.This effect will be deeply discussed in Sec.III.The shape of the conﬁningﬁeld alsoﬁxes the symme-try of the problem.One can use spherical or axially sym-metric traps,for instance.Theﬁrst experiments on ru-bidium and sodium were carried out with axial symmetry.In this case,one can deﬁne an axial coordi-nate z and a radial coordinate rЌϭ(x2ϩy2)1/2and the corresponding frequencies,␻z and␻Ќϭ␻xϭ␻y.The ra-tio between the axial and radial frequencies,␭ϭ␻z/␻Ќ,ﬁxes the asymmetry of the trap.For␭Ͻ1the trap is cigar shaped while for␭Ͼ1is disk shaped.In terms of␭the ground state Eq.(3)for noninteracting bosons can be rewritten as␸0͑r͒ϭ␭1/43/4aЌ3/2expͫϪ12aЌ2͑rЌ2ϩ␭z2͒ͬ.(6)Here aЌϭ(ប/m␻Ќ)1/2is the harmonic-oscillator length in the xy plane and,since␻Ќϭ␭Ϫ1/3␻ho,one has also aЌϭ␭1/6a ho.The choice of an axially symmetric trap has proven useful for providing further evidence of Bose-Einstein condensation from the analysis of the momentum distri-bution.To understand this point,let us take the Four-ier transform of the wave function Eq.(6):␸˜0(p)ϰexp͓ϪaЌ2(pЌ2ϩ␭Ϫ1p z2)/2ប2͔.From this one can calculate the average axial and radial widths.Their ratio,ͱ͗z2͗͘Ќ2͘ϭͱ␭,(7) isﬁxed by the asymmetry parameter of the trap.Thus the shape of the expanded cloud in the xz plane is an ellipse,the ratio between the two axis(aspect ratio)be-ing equal toͱ␭.If the particles,instead of being in the lowest state(condensate),were thermally distributed among many eigenstates at higher energy,their distribu-tion function would be isotropic in momentum space, according to the equipartition principle,and the aspect ratio would be equal to1.Indeed,the occurrence of anisotropy in the condensate peak has been interpreted from the very beginning as an important signature of BEC(Anderson et al.,1995;Davis et al.,1995;Mewes et al.,1996a).In the case of the experiment at the Joint Institute for Laboratory Astrophysics(JILA)in Boul-der,the trap is disk-shaped with␭ϭͱ8.Theﬁrst mea-sured value of the aspect ratio was about50%larger than the prediction,ͱ␭,of the noninteracting model (Anderson et al.,1995).Of course,a quantitative com-parison can be obtained only including the atom-atom interaction,which affects the dynamics of the expansion (Holland and Cooper,1996;Dalfovo and Stringari,1996; Holland et al.,1997;Dalfovo,Minniti,Stringari,and Pi-taevskii,1997).However,the noninteracting model al-ready points out this interesting effect due to anisotropy.B.Trapped bosons atﬁnite temperature:thermodynamic limitAt temperature T the total number of particles is given,in the grand-canonical ensemble,by the sumNϭ͚n x,n y,n z͕exp͓␤͑␧n x n y n zϪ␮͔͒Ϫ1͖Ϫ1,(8) while the total energy is given byEϭ͚n x,n y,n z␧n x n y n z͕exp͓␤͑␧n x n y n zϪ␮͔͒Ϫ1͖Ϫ1,(9) where␮is the chemical potential and␤ϭ(k B T)Ϫ1.Be-low a given temperature the population of the lowest state becomes macroscopic and this corresponds to the onset of Bose-Einstein condensation.The calculation of the critical temperature,the fraction of particles in the lowest state(condensate fraction),and the other ther-modynamic quantities,starts from Eqs.(8)and(9)with the appropriate spectrum␧nxn y n z(de Groot,Hooman, and Ten Seldam,1950;Bagnato,Pritchard,and Klepp-ner,1987).Indeed the statistical mechanics of these trapped gases is less trivial than expected atﬁrst sight. Several interesting problems arise from the fact that these systems have aﬁnite size and are inhomogeneous. For example,the usual deﬁnition of thermodynamic limit(increasing N and volume with the average density kept constant)is not appropriate for trapped gases. Moreover,the traps can be made very anisotropic, reaching the limit of quasi-two-dimensional and quasi-one-dimensional systems,so that interesting effects of reduced dimensionality can be also investigated.As in the case of a uniform Bose gas,it is convenient to separate out the lowest eigenvalue␧000from the sum (8)and call N0the number of particles in this state.This number can be macroscopic,i.e.,of the order of N,when the chemical potential becomes equal to the energy of the lowest state,␮→␮cϭ32ប␻¯,(10)where␻¯ϭ(␻xϩ␻yϩ␻z)/3is the arithmetic average of the trapping frequencies.Inserting this value in the rest of the sum,one can writeNϪN0ϭ͚n x,n y,n z 01exp͓␤ប͑␻x n xϩ␻y n yϩ␻z n z͔͒Ϫ1.(11) In order to evaluate this sum explicitly,one usually as-sumes that the level spacing becomes smaller and smaller when N→ϱ,so that the sum can be replaced by an integral:NϪN0ϭ͵0ϱdn x dn y dn zexp͓␤ប͑␻x n xϩ␻y n yϩ␻z n z͔͒Ϫ1.(12) This assumption corresponds to a semiclassical descrip-tion of the excited states.Its validity implies that the relevant excitation energies,contributing to the sum (11),are much larger than the level spacingﬁxed by the468Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999。

tpo40三篇托福阅读TOEFL原文译文题目答案译文背景知识

tpo40三篇托福阅读TOEFL原文译文题目答案译文背景知识阅读-1 (2)原文 (2)译文 (5)题目 (8)答案 (17)背景知识 (17)阅读-2 (20)原文 (20)译文 (23)题目 (25)答案 (35)背景知识 (35)阅读-3 (38)原文 (38)译文 (41)题目 (44)答案 (53)背景知识 (54)阅读-1原文Ancient Athens①One of the most important changes in Greece during the period from 800 B.C. to 500 B.C. was the rise of the polis, or city-state, and each polis developed a system of government that was appropriate to its circumstances. The problems that were faced and solved in Athens were the sharing of political power between the established aristocracy and the emerging other classes, and the adjustment of aristocratic ways of life to the ways of life of the new polis. It was the harmonious blending of all of these elements that was to produce the classical culture of Athens.②Entering the polis age, Athens had the traditional institutions of other Greek protodemocratic states: an assembly of adult males, an aristocratic council, and annually elected officials. Within this traditional framework the Athenians, between 600 B.C. and 450 B.C., evolved what Greeks regarded as a fully fledged democratic constitution, though the right to vote was given to fewer groups of people than is seen in modern times.③The first steps toward change were taken by Solon in 594 B.C., when he broke the aristocracy's stranglehold on elected offices by establishing wealth rather than birth as the basis of office holding, abolishing the economic obligations of ordinary Athenians to the aristocracy, and allowing the assembly (of which all citizens were equal members) to overrule the decisions of local courts in certain cases. The strength of the Athenian aristocracy was further weakened during the rest of the century by the rise of a type of government known as a tyranny, which is a form of interim rule by a popular strongman (not rule by a ruthless dictator as the modern use of the term suggests to us). The Peisistratids, as the succession of tyrants were called (after the founder of the dynasty, Peisistratos), strengthened Athenian central administration at the expense of the aristocracy by appointing judges throughout the region, producing Athens’ first national coinage, and adding and embellishing festivals that tended to focus attention on Athens rather than on local villages of the surrounding region. By the end of the century, the time was ripe for more change: the tyrants were driven out, and in 508 B.C. a new reformer, Cleisthenes, gave final form to the developments reducing aristocratic control already under way.④Cleisthenes' principal contribution to the creation of democracy at Athens was to complete the long process of weakening family and clanstructures, especially among the aristocrats, and to set in their place locality-based corporations called demes, which became the point of entry for all civic and most religious life in Athens. Out of the demes were created 10 artificial tribes of roughly equal population. From the demes, by either election or selection, came 500 members of a new council, 6,000 jurors for the courts, 10 generals, and hundreds of commissioners. The assembly was sovereign in all matters but in practice delegated its power to subordinate bodies such as the council, which prepared the agenda for the meetings of the assembly, and courts, which took care of most judicial matters. Various committees acted as an executive branch, implementing policies of the assembly and supervising, for instance, the food and water supplies and public buildings. This wide-scale participation by the citizenry in the government distinguished the democratic form of the Athenian polis from other less liberal forms.⑤The effect of Cleisthenes’ reforms was to establish the superiority of the Athenian community as a whole over local institutions without destroying them. National politics rather than local or deme politics became the focal point. At the same time, entry into national politics began at the deme level and gave local loyalty a new focus: Athens itself. Over the next two centuries the implications of Cleisthenes’ reforms were fully exploited.⑥During the fifth century B.C. the council of 500 was extremely influential in shaping policy. In the next century, however, it was the mature assembly that took on decision-making responsibility. By any measure other than that of the aristocrats, who had been upstaged by the supposedly inferior "people", the Athenian democracy was a stunning success. Never before, or since, have so many people been involved in the serious business of self-governance. It was precisely this opportunity to participate in public life that provided a stimulus for the brilliant unfolding of classical Greek culture.译文古雅典①在公元前800年到公元前500年期间，希腊最重要的变化之一是城邦的崛起，并且每个城邦都发展了适合其情况的政府体系。

Giant d 33 in (K,Na)(Nb,Sb)O3-(Bi,Na,K, Li)ZrO3 based lead-free piezoelectrics with high T c

Giant d 33 in (K,Na)(Nb,Sb)O3-(Bi,Na,K, Li)ZrO3 based lead-free piezoelectrics with high T cXiaojing Cheng, Jiagang Wu, Xiaopeng Wang, Binyu Zhang, Jianguo Zhu, Dingquan Xiao, Xiangjian Wang, and Xiaojie LouCitation: Applied Physics Letters 103, 052906 (2013); doi: 10.1063/1.4817517View online: /10.1063/1.4817517View Table of Contents: /content/aip/journal/apl/103/5?ver=pdfcovPublished by the AIP PublishingArticles you may be interested inMorphotropic phase boundary and electric properties in (1x)Bi0.5Na0.5TiO3-xBiCoO3 lead-free piezoelectric ceramicsJ. Appl. Phys. 111, 124113 (2012); 10.1063/1.4730770A monoclinic-tetragonal ferroelectric phase transition in lead-free (K0.5Na0.5)NbO3-x%LiNbO3 solid solutionJ. Appl. Phys. 111, 103503 (2012); 10.1063/1.4716027Determination of depolarization temperature of (Bi1/2Na1/2)TiO3-based lead-free piezoceramicsJ. Appl. Phys. 110, 094108 (2011); 10.1063/1.3660253High piezoelectric activity in (Na,K)NbO3 based lead-free piezoelectric ceramics: Contribution of nanodomains Appl. Phys. Lett. 99, 062901 (2011); 10.1063/1.3624704Piezoelectric and ferroelectric properties of [ Bi 0.5 ( Na 1 x y K x Li y ) 0.5 ] Ti O 3 lead-free piezoelectric ceramicsAppl. Phys. Lett. 88, 062901 (2006); 10.1063/1.2171799Giant d 33in (K,Na)(Nb,Sb)O 3-(Bi,Na,K,Li)ZrO 3based lead-free piezoelectrics with high T cXiaojing Cheng,1Jiagang Wu,1,a)Xiaopeng Wang,1Binyu Zhang,1Jianguo Zhu,1Dingquan Xiao,1,b)Xiangjian Wang,2and Xiaojie Lou 21Department of Materials Science,Sichuan University,Chengdu 610064,People’s Republic of China 2Multi-disciplinary Materials Research Center,Frontier Institute of Science and Technology,Xi’an Jiaotong University,Xi’an 710054,People’s Republic of China(Received 12May 2013;accepted 19July 2013;published online 1August 2013)In this work,a lead-free piezoelectric system based on (1Àx )(K 0.48Na 0.52)(Nb 0.95Sb 0.05)O 3-x Bi 0.5(Na 0.7K 0.2Li 0.1)0.5ZrO 3[(1Àx )KNNS-x BNKLZ]is developed,and a rhombohedral-tetragonal phase boundary is constructed in this system.The relationship betweenboundaryand the piezoelectricproperties ofthe (1Àx)KNNS-xBNKLZ ceramicsis The coexistence of a tetragonal phase and a rhombohedral phase is identiﬁed in the range of 0.03<x <0.05.For such composition,the ceramics show giant d 33of $380pC/N,high T c of $290 C,and a good thermal-depolarization behavior (d 33>300pC/N)of $210 C.We believe that the (1Àx )KNNS-x BNKLZ system is very promising for lead-free piezoelectricapplications.VC 2013AIP Publishing LLC .[/10.1063/1.4817517]The construction and modiﬁcation of the phase boundaryhave long been one of the hottest research topics in the ﬁeld of piezoelectric materials.1–9Recently,considerable attention has been paid to the modiﬁcation of the phase boundary in (K,Na)NbO 3(KNN)-based lead-free ceramics with the aim of obtaining excellent piezoelectric properties comparable to toxic lead-containing Pb(Zr,Ti)O 3(PZT).3–9In the phase boundaries have been used to enhance the piezoelectric properties of KNN-based ceramics,3–19that is,the orthorhombic and tetragonal (O-T)phase boundary and the rhombohedral-orthorhombic (R-O)phase boundary.During the past years,more attention has been given to the construction of an O-T phase boundary in KNN-based ceramics.3–7,10–12,17–19It was found that after such phase con-struction,the piezoelectric constant (d 33)of the KNN-based ceramics has been greatly improved.The R-O phase bound-ary has not been designed in KNN-based ceramics until very recently by a few research groups using the method of doping certain types of 14,15The additives play an impor-tant role in R-O phase transition point from low temperatures to room temperature under pres-sure.However,these KNN-based ceramics with phase boundary show poorer d 33than those with an O-T phase boundary.3–7,10–15,17–19On the other hand,the enhancement in d 33of KNN-based ceramics was often achieved by the sac-riﬁce and signiﬁcant reduction of their T c point.3,8,9From this regard,it is necessary to develop a type of KNN-based ceramics with both high d 33and high Curie temperature (T c ).A phase boundary,i.e.,the tetragonal and rhombohedral (T-R)phases coexistence,similar to the situation in PZT,has been successfully constructed in order to improve the piezo-electric behavior (d 33>300pC/N)of KNN-based ceramics.We ﬁnd that the ceramics with the composition lying in the T-R phase boundary show dramatically enhanced d 33andsigniﬁcantly improved T c ,both of which are superior to those located at the O-T and O-R phase boundaries.It is well known that for the O-T and O-R phase boundary,the low T c of <200 C hinders their practical applications,8,9as shown in Table I .In this work,we introduce a additive of Bi 0.5(Na 0.7K 0.2Li 0.1)0.5ZrO 3to (K 0.48Na 0.52)(Nb 0.95Sb 0.05)O 3for improving its d 33without signiﬁcantly decreasing its T c by constructing a tetragonal and rhombohedral phase boundary at room temperature.As shown in Fig.1(d),after adding [Bi 0.5(Na 0.7K 0.2Li 0.1)0.5]2þ,the O-T phase boundary can be to room temperature.The aim of adding Zr 4þis to the R-O phase boundary of (K 0.48Na 0.52)(Nb 0.95Sb 0.05)O 3to room temperature.small amount of Nb 5þby Sb 5þcould stabilize the of the T-R phase at/near room temperature.Adding a small amount of Li þcould increase T c and decrease both T O-T and the sintering temperature.4,20Finally,maintaining the ratio of Na,K,and Li in [Bi 0.5(Na 0.7K 0.2Li 0.1)0.5]2þto be 7:2:1may help enhance the piezoelectric properties of the sample according to a previous study by Lin et al.21Then,(1Àx )(K 0.48Na 0.52)(Nb 0.95Sb 0.05)O 3-x Bi 0.5(Na 0.7K 0.2Li 0.1)0.5ZrO 3[(1Àx )KNNS-x BNKLZ]lead-free ceramics were pre-pared by the conventional solid reaction and ceramic proc-essing methods.Indeed,we found that a R-T phase boundary can be constructed in such a material system.A large d 33of $380pC/N and a high T c of $290 C have been obtained for the ceramics with the composition lying in the R-T phase boundary.Finally,the underlying physical mechanism has been investigated and discussed.The (1Àx )KNNS-x BNKLZ (x ¼0$0.07)ceramics were prepared by normal sintering method.The raw materi-als used in this work are K 2CO 3,Na 2CO 3,Nb 2O 5,Sb 2O 3,Bi 2O 3,Li 2CO 3,and ZrO 2.These powders were weighed according to the chemical formula of (1Àx )KNNS-x BNKLZ (x ¼0$0.07),ball mixed for 24h in and then cal-850 C for 6h in air.These were a polyvinyl alcohol (PVA)anda)Author to whom correspondence should be addressed.Electronic addresses:msewujg@ and wujiagang0208@ b)Email:nic0402@0003-6951/2013/103(5)/052906/5/$30.00VC 2013AIP Publishing LLC 103,052906-1APPLIED PHYSICS LETTERS 103,052906(2013)pressed into a disk sample with a diameter of $1.0cm and a thickness of $1.0mm.All disk samples were sintered at 1100–1150 C for 3h in air after burning out the PVA binder.The silver paste on both sides of thesewasﬁred at700C for10minand usedasfor electri-cal measurements.All samples have been a tempera-ture range of room temperature $150 C in a silicone oil bath under a dc ﬁeld of 3.0–4.0kV/mm for 20min.X-ray diffraction (XRD)(Bruker D8Advanced XRD,Bruker AXS,Inc.,Madison,WI,CuKa)was carried out to measure the structural properties of the samples in the 2h scan range of 20 –60 .Scanning electron microscopy (SEM)(Philips,XL30)was employed to study ogy of these sintered samples.The and tion factors of the sintered samples LCR analyzer (HP 4980,Agilent,USA.)with a varied tem-perature between À150–200 C and 28–550 P-E hysteresis loops were tested at 10Hz by a Workstation (USA).Their piezoelectric d 33was measured using a quasi-static d 33meter (ZJ-3A,Institute of Acoustics,Chinese Academy of Sciences,China),and an im-(Solartron Gain Phase Analyzer)was to characterize their dielectric properties.The res-method was used to characterize the k p to the IEEE standards by using an impedance ana-lyzer (HP 4294A,Agilent,USA),and the quality factor (Q m )was calculated using of Q m ¼f a 2(f a 2Àf r 2)/2p f r Z m C T ,where the f r and f a are,respec-tively,the resonance frequency (Hz)and the antiresonance frequency (Hz),the Z m is the minimum impedance at f r (ohm),and the C T is mechanically free at 1kHz.Figure 1(a)shows the patterns of the (1Àx )KNNS-x BNKLZ ceramics in 2h ¼20o –60 .A pure peroviskite phase is observed in all the samples,with no secondary phases.To further characterize their phase structure,we conducted the XRD scans in the 2h range of 44 –47 ,as shown in Fig.1(b).For x <0.02,an orthorhom-bic phase is evident.The coexistence of the orthorhombic phase and the tetragonal phase appears for the composition of 0.02 x 0.03.The ceramics with the composition of x !0.05show a rhombohedral phase.Figure 1(c)shows the XRD pattern of the (1Àx )KNNS-x BNKLZ ceramics with x ¼0.04,along with the simulation using the Lorentz method for identifying the proportion of the R and T phases.From Fig.1(c),the coexistence of the tetragonal and rhombohedral phases can be observed in such a ceramic.After detailed analysis,the composition range for the coexistence of a tet-ragonal and rhombohedral phase can be identiﬁed to be 0.03<x <0.05.The temperature-dependent capacitance of these ceramics were carried out in the decreasing temperature from 200 C to À150 C to identify the phase evolution of T R-O and T O-T ,as is shown in Figs.2(a)–2(d).T R-O and T O-T could be clearly observed for the the compositions of x ¼0and 0.03.T R-O and T O-T as the BNKLZ content increases to x ¼0.04,leading to of the R-T phase bound-ary at room temperature.To check the effect of the BNKLZ content on T c and the phase evolution,the temperature de-pendent dielectric constant (e r )of the (1Àx )KNNS-x BNKLZ ceramics was measured at 100kHz in the temperature range of 28–550 C,as shown in Fig.3(a).Fig.3(b)shows T c as a function of x ,according to Fig.3(a).The T c curve shows a plateau for x 0.04,and then drops dramatically for x >0.04.Therefore,the composition with x ¼0.04shows a relatively high T c of $290 C,which is much higher than other KNN-based ceramics with R-T.8,9Figure 3(b)also shows the phase diagram of the (1Àx )KNNS-x BNKLZ ceramics according to the temperature-dependent dielectric properties shown inTABLE I.Parameters of KNN-based ceramics and PZT4with a large d positiond 33(pC/N)k p T c (o C)References (Na 0.52K 0.40)(Nb 0.84Sb 0.08)O 3-(0.08-x )LiTaO 3-x BaZrO 3(x ¼0.025)3650.45$18080.94(K 0.4Àx Na 0.6Ba x Nb 1Àx Zr x )O 3–0.06LiSbO 3(x ¼0.06)3440.33$1809(K 0.44Na 0.52Li 0.04)(Nb 0.84Ta 0.10Sb 0.06)O 34160.61$2533PZT44100.60$2503(1Àx )(K 0.48Na 0.52)(Nb 0.95Sb 0.05)O 3-x Bi 0.5(Na 0.7K 0.2Li 0.1)0.5ZrO 3(x ¼0.04)3800.46$290This workFIG.1.XRD patterns of (1Àx )KNNS-x BNKLZ ceramics in the 2h range of (a)20 –60 and (b)44 –47 ,(c)expanded XRD pattern of the ceramic with x ¼0.04,and (d)design diagram for the R-T phase boundary in (1Àx )KNNS-x BNKLZ system.Figs.2and 3(a).From Figure 3(b),one sees that the R-T phase boundary is clearly formed in the ceramic with x ¼0.04at room temperature.Therefore,using XRD and the temperature-dependent dielectric measurements,we conﬁrm the formation of the R-T phase boundary in the ceramic with x ¼0.04.Figures 4(a)–4(d)show the surface morphologies of the (1Àx )KNNS-x BNKLZ ceramics for different BNKLZ con-tents.The grain size gradually increases with increasing BNKLZ content (x 0.04)and then drops dramatically with further increasing BNKLZ content (x ¼0.07).Also,the ceramics with x 0.04show an irregular distribution of grain sizes,that is,bigger grains are surrounded by smaller ones.The reason for the highly distributed grain size in these ceramics is probably due to the following factors:(1)For low BNKLZ contents,the formation of a liquid phase caused by Bi 3þcould promote the grain growth of the ceramics,giving rise to the increase in grain size.22(2)For high BNKLZ con-tents,the excessive BNKLZ couldatgrain bounda-ries andthe Zr 4þgrowth during sintering process in a decrease in grain size.Moreover,the ceramics with x <0.07show a dense micro-structure,while the one with x ¼0.07is very porous.Figure 5(a)shows the P-E loops of the (1Àx )KNNS-x BNKLZ ceramics,measured at 10Hz and room tempera-ture.All the samples show typical square-like ferroelectric P-E loops except for the one with x ¼0.07,due to the involvement of a rhombohedral phase.8The plots of 2P r and 2E c vs.the BNKLZ content are shown in the inset of Fig.5(a).From this ﬁgure,one sees that 2P r gradually drops with increasing BNKLZ content,and 2E c slightly ﬂuctuates for x <0.06and then drops for x !0.06.Figure 5(b)shows e r ,tan d ,d 33,k p ,and Q m as a function of the BNKLZ con-tent,with the aim of investigating the change in their dielectric and piezoelectric properties around the morpho-tropic phase boundary (MPB).We see that e r reaches the maximum in the R-T phase boundary region,and tan d slightly ﬂuctuates with the BNKLZ content.A larger tan d is observed in the ceramic with x ¼0.07because of its poor microstructure.Interestingly,we ﬁnd that d 33reaches the maximum value of $380pC/N and k p reaches the maxi-mum value of $0.46for x ¼0.04.However,Q m gradually decreases as the BNKLZ content increases.Therefore,we successfully obtained the (1Àx )KNNS-x BNKLZ ceramics with x ¼0.04,showing an dramatically enhanced dielectric and piezoelectric properties with e r $2180,tan d $3.5%,d 33$380pC/N,and k p $0.46.Figure 6(a)shows the thermal stability of d 33and k p in the (1Àx )KNNS-x BNKLZ ceramics with x ¼0.04,measured at room temperature after the samples were at vary-ing temperatures (T a )for 120min.From this one sees that d 33and k p slightly decrease with increasing T aFIG. 2.Temperature dependence of the capacitance of (1Àx )KNNS-x BNKLZ ceramics with (a)x ¼0,(b)x ¼0.02,(c)x ¼0.04,and (d)x ¼0.06,measured in the temperature range of À150–200 C and 10kHz.FIG.3.(a)Temperature dependence of the dielectric constant of (1Àx )KNNS-x BNKLZ ceramics in the temperature range of 28–550 C and (b)phase dia-gram of (1Àx )KNNS-x BNKLZ ceramics.(T a 240 C)and suddenly drop with further increasing T a (T a >240 C)due to the fact that T a is approaching T c .Nevertheless,a high d 33(>300pC/N)maintains in the T a range of 25–210 C,demonstrating a good thermal stability in such a ceramic.In this work,wethe ceramic with x ¼0.04showing a high d 33of $which is much larger than those KNN-based ceramics with O-R and O-T 3–7,10–15,17–19and is even comparable to those KNN-based ceramics with R-T or textured morphologies.3,8,9,24Moreover,it is of great im-portance that the ceramic with x ¼0.04also shows a much higher T c ($290 C)than the textured KNN-based ceramics with the same phase boundary,or PZT4,3,8,9as shown inTable I .The higher T c beneﬁts the industrial applications in a wide temperature range.The physical mechanisms that may be responsible for the enhanced piezoelectric properties in the ceramic with x ¼0.04are as follows:First,the existence of the R-T phase boundary in the (1Àx )KNNS-x BNKLZ ceramics with x ¼0.04may largely promote its d 33because of more ori-entation states of spontaneous polarization.8,9Second,it is well known that a much higher e r P r generally corresponds to a much larger d 33.9,25Therefore,e r P r is used to illuminate the physical mechanism for the enhancement in d 33in these piezo-electric materials.Figure 6(b)plots e r P r and d 33of the (1Àx )KNNS-x BNKLZ ceramics as a function of the BNKLZ content.Indeed,we see that the proﬁle for e r P r and the one for d 33show similar trend as the BNKLZ content increases;both e r P r and d 33reach the maximum values for the ceramic with x ¼0.04.Therefore,it is reasonable to partly attribute the sig-niﬁcant enhancement in d 33of the ceramic with x ¼0.04to its highest e r P r .9,25Additionally,it is well known that the MPB of a speciﬁc material depends only on its composition,while the polymorphic phase boundary (PPB)is dependent of not only its composition but also the temperature.5–7,24–29In this work,the R-T phase transition makes the dielectric property peak at corresponding speciﬁc temperature,as shown in Fig.2(c).Such a phenomenon is similar to those observed in Pb(Mg 1/3Nb 2/3)O 3-PbTiO 3(PMN-PT),30,31PbMg 1/3Nb 2/3O 3-PbZrO 3-PbTiO 3(PMN-PZT),32and PbIn 1/2Nb 1/2O 3-PbMg 1/3N b2/3O 3-PbTiO 3(PIN-PMN-PT)33,34single crystals with rombohedral-tetragonal MPB.Since the R-T phase boundary of PMN-PT,PMN-PZT,and PIN-PMN-PT single crystals is well known to be an MPB,rather than a PPB,30–34we argue that the (1Àx )KNNS-x BNKLZ ceramics studied in this work may also have a morphotropic character.However,it has been reportedFIG.4.Surface morphologies of (1Àx )KNNS-x BNKLZ ceramics with dif-ferent BNKLZ content:(a)x ¼0,(b)x ¼0.02,(c)x ¼0.04,and (d)x ¼0.07.FIG.5.P-E loops and (b)E r ,tan d ,d 33,k p ,and Q m values of (1Àx )KNNS-x BNKLZ ceramics as a function of BNKLZ content,and the inset in (a)is 2P r and 2E c values vs.BNKLZ content.FIG.6.(a)Thermal stability of d 33and k p values of (1Àx )KNNS-x BNKLZ ceramics with x ¼0.04and (b)their E r P r and d 33vs.BNKLZ content.that the piezoelectric properties of KNN-based ceramics show strong temperature dependence.5–7Therefore,the real nature of the phase boundary in the(1Àx)KNNS-x BNKLZ ceramics warrants systematic investigations in the future.In conclusion,(1Àx)(K0.48Na0.52)(Nb0.95Sb0.05)O3-x Bi0.5 (Na0.7K0.2Li0.1)0.5ZrO3ceramics were prepared using the conventional solid-state method.These ceramics with0.03<x<0.05show a coexistence of a tetragonal phase anda rhombohedral phase at room temperature.Correspondingly, an enhanced piezoelectric behavior is observed in the ceramics with compositions lying in the phase boundary zone.The ceramic with x¼0.04shows an optimum electrical properties with d33$380pC/N,k p$0.46,e r$2180,tan d$3.5%,2P r$29.3l C/cm2,and2E c$20.5kV/cm because of the involvement of the R-T phase boundary and the enhanced e r P r in this speciﬁc composition.Furthermore,both a large d33($380pC/N)and a high T c($290 C),superior to other rhombohedral-tetragonal KNN-based ceramics,are achieved in a single ceramic with x¼0.04.We believe that the(1Àx)KNNS-x BNKLZ ceramics studied in this work may shed light on the methodologies of designing high-performance KNN-based lead-free piezoelectric materials and is therefore very promising for lead-free piezoelectric applications.Authors gratefully acknowledge the supports of the National Science Foundation of China(NSFC Nos. 51102173and51272164),the introduction of talent start funds of Sichuan University(2082204144033),the Fundamental Research Funds for the Central Universities (2012SCU04A01),and the National Students’Innovation and Entrepreneurship Training Program(201210610005).1W.F.Liu and X.Ren,Phys.Rev.Lett.103,257602(2009).2D.Damjanovic,Appl.Phys.Lett.97,062906(2010).3Y.Saito,H.Takao,T.Tani,T.Nonoyama,K.Takatori,T.Homma,T. Nagaya,and M.Nakamura,Nature432,84(2004).4Y.P.Guo,K.Kakimoto,and H.Ohsato,Appl.Phys.Lett.85(18),4121 (2004).5T.R.Shrout and S.J.Zhang,J.Electroceram.19(1),113(2007).6E.Hollenstein,M.Davis,D.Damjanovic,and N.Setter,Appl.Phys.Lett. 87,182905(2005).7J.R€o del,W.Jo,K.T.P.Seifert,E.M.Anton,T.Granzow,and D. Damjanovic,J.Am.Ceram.Soc.92,1153(2009).8R.Zuo and J.Fu,J.Am.Ceram.Soc.94,1467(2011).9W.F.Liang,W.J.Wu,D.Q.Xiao,J.G.Zhu,and J.G.Wu,J.Mater.Sci. 46,6871(2011).10B.Q.Ming,J.F.Wang,P.Qi,and G.Z.Zang,J.Appl.Phys.101(5), 054103(2007).11J.G.Wu,D.Q.Xiao,Y.Y.Wang,J.G.Zhu,L.Wu,and Y.H.Jiang, Appl.Phys.Lett.91,252907(2007).12K.Wang and J.F.Li,Adv.Funct.Mater.20,1924(2010).13S.J.Zhang,R.Xia,and T.R.Shrout,Appl.Phys.Lett.91,132913 (2007).14R.P.Wang,H.Bando,T.Katsumata,Y.Inaguma,H.Taniguchi,and M. Itoh,Phys.Status Solidi(RRL)3,142(2009).15R.Z.Zuo,J.Fu,D.Y.Lv,and Y.Liu,J.Am.Ceram.Soc.93(9),2783 (2010).16M.Matsubara,K.Kikuta,and S.Hirano,J.Appl.Phys.97,114105 (2005).17E.K.Akdogan,K.Kerman,M.Abazari,and A.Safari,Appl.Phys.Lett. 92,112908(2008).18Y.P.Guo,K.Kakimoto,and H.Ohsato,Mater.Lett.59,241(2005).19Y.F.Chang,Z.P.Yang,D.F.Ma,Z.H.Liu,and Z.L.Wang,J.Appl. Phys.104,024109(2008).20N.Klein,E.Hollenstein,D.Damjanovic,H.J.Trodahl1,N.Setter,and M. Kuball,J.Appl.Phys.102,014112(2007).21D.Lin,D.Xiao,J.Zhu,and P.Yu,Appl.Phys.Lett.88,062901(2006). 22H.Du,F.Luo,S.Qu,Z.Pei,D.Zhu,and W.Zhou,J.Appl.Phys.102, 054102(2007).23B.Malic,J.Bernard,A.Bencan,and M.Kosec,J.Eur.Ceram.Soc.28, 1191(2008).24B.Jaffe,W.R.Cook,and H.Jaffe,Piezoelectric Ceramics(Academic Press,London,1971).25S.E.Park and T.R.Shrout,IEEE Trans.Ultrason.Ferroelectr.Freq. Control44,1140(1997).26D.Damjanovic,J.Am.Ceram.Soc.88,2663(2005).27M.Ahtee and A.M.Glazer,Acta Crystallogr.,Sect.A:Found. Crystallogr.A32,434(1976).28S.E.Park,S.Wada,L.E.Cross,and T.R.Shrout,J.Appl.Phys.86,2746 (1999).29M.Matsubara,T.Yamaguchi,K.Kikuta,and S.Hirano,Jpn.J.Appl. Phys.,Part144,6136(2005).30X.Zhao,B.Fang,H.Cao,Y.Guo,and H.Luo,Mater.Sci.Eng.,B96, 254(2002).31F.Li,S.Zhang,Z.Xu,X.Wei,J.Luo,and T.R.Shrout,J.Appl.Phys. 108,034106(2010).32S.Zhang,S.M.Lee,D.H.Kim,H.Y.Lee,and T.R.Shrout,Appl.Phys. Lett.90,232911(2007).33X.Liu,S.Zhang,J.Luo,T.R.Shrout,and W.Cao,J.Appl.Phys.106, 074112(2009).34S.Zhang,J.Luo,W.Hackenberger,and T.R.Shrout,J.Appl.Phys.104, 064106(2008).。

Metal-insulator transition in 2D resistance in the critical region

a r X i v :c o n d -m a t /0003032v 2 [c o n d -m a t .m e s -h a l l ] 20 A p r 2000Metal-insulator transition in 2D:resistance in the critical regionB.L.Altshuler 1,2,D.L.Maslov,3and V.M.Pudalov 41)NEC Research Institute,4Independence Way,Princeton,NJ 085402)Physics Department,Princeton University,Princeton,NJ 085443)Department of Physics,University of Florida P.O.Box 118440,Gainesville,Florida 32611-84404)P.N.Lebedev Physics Institute,Russian Academy of SciencesLeninsky Prospect 53,Moscow 117924,Russia(February 1,2008)The goal of this paper is to highlight several issues which are most crucial for the understanding of the “metal-insulator transition”in two dimensions.We discuss some common problems in in-terpreting experimental results on high mobility Si MOSFETs.We analyze concepts and methods used to determine the critical density of electrons at the metal-insulator transition.In particular,we discuss the origin of the temperature dependence of the resistivity and reasons for this dependence to ﬂatten out at some electron density in the vicinity of the metal-insulator transition.This ﬂattening has recently been proposed to indicate a true quantum phase transition.We suggest an alternative interpretation of this result and demonstrate the consistency of our proposition with the experimen-tal data.One of the main questions,which arise in connection with the transition,is whether or not the metallic state is qualitatively distinct from a conventional disordered Fermi liquid.We analyze the arguments in favor of both aﬃrmative and negative answers to this question and conclude that the experimental results accumulated up-to-date do not provide convincing evidence for the new state of matter characterized by a metallic-like residual conductivity.We also discuss in details the measurement and control of the electron temperature;these issues are crucial for interpreting the low-temperature experimental data.PACS numbers:71.30.+h,72.15Rn,73.40.QvI.INTRODUCTIONAfter several years of intensive experimental and the-oretical eﬀorts(see,e.g.,Ref.[1]for an extensive bib-liography),even the basic features of the phenomenon known as the“metal-insulator transition in two dimen-sions”(2D MIT)remain to be the subjects of ongoing discussions.Is this phenomenon a true“quantum phase transition”(QPT)[2]or can it be understood in terms of conventional physics of disordered conductors[3,4]? This question is at the heart of the whole discussion. Recently,we wrote a paper[5]in favor of the second possibility.In particular,we argued thati)in the“metallic phase”,2D systems seem to behaveas a quite conventional disordered metal ratherthan a distinctly new state of matter and ii)it is possible to explain the anomalous behavior of the resistivity,ρ(T),by an interplay of two temper-ature dependences:the one given by a metallic-like,quasiclassical(Drude)resistivity[6],ρd(T),andthe other one arising from quantum localization. We do realize that ii)represents a rather naive ap-proach,at least because it does not fully take into account the Coulomb interaction between electrons,whereas the most pronounced eﬀects have been observed in systems with a priori strong electron-electron correlations.Nev-ertheless,this approach allows one to describe qualita-tively,and even semi-quantitatively,most of the results on electron transport both in metallic phase and near the transition point.This suggests that it is a good starting point for developing an adequate theoretical understand-ing of the MIT in two dimensions.Conclusion i)is based on the analysis of existing ex-perimental data,as summarized in Refs.[5,7].It has been further supported by recent experimental papers[ 1,8–10],which showed that the transport properties of Si/Ge,Si MOSFET and p-GaAs structures in the metal-lic regime can be successfully interpreted in conventional terms.Authors of Refs.[9,10]have also identiﬁed the contribution of electron-electron interactions to the resis-tance(via measuring the temperature-dependent part of the Hall resistance)and found that it remains quite small even when parameter r s is rather large.There is also a number of recent theoretical papers[11–14,7]which, although diﬀering in a particular mechanism for the T-dependence of the Drude resistivity,share the general spirit of propositions i)and ii).An alternative point of view,expressed,for example, in Refs.[15–17],is that the observed MIT-like behavior indicates a true quantum phase transition between an insulator and a novel metallic phase.Motivated by the importance of this controversy,i.e.,QPT versus proposi-tions i)and ii),for theﬁeld of quantum transport in2D and its possible relevance for other realizations of quan-tum phase transitions,we decided to analyze in detail the arguments on both sides of the QPT question.In this paper,we discuss recent and some of the previously pub-lished experimental results on the resistivity of Si MOS-FETs in the vicinity of the MIT,along with some of the theoretical interpretations of these results.We conclude that that there is no convincing experimental evidence for the QPT in the existing data.As one of the main questions in theﬁeld is the behavior of a2D system in the limit T→0,we begin our analysis with a short summary of recent results by Kravchenko and Klapwijk(KK)[15],who measured the temperature dependence of the resistivity down to a bath temperature as low as T bath=35mK.This paper reports measure-ments on a single Si MOSFET sample with peak mobil-ityµpeak=27,000cm2/Vs in the range from35mK to 1.2K atﬁve diﬀerent electron concentrations.Digitized data from Ref.[15]is shown in Fig.1(curves1-5).In Ref.[15],the following two points are emphasized:1)For n=n1=6.85×1010cm−2and n=n2=7.17×1010cm−2(curves1and2),the resistivitydecreases as temperature increases(dρ/dT<0);for n=n4=7.57×1010cm−2and n=n5=7.85×1010cm−2(curves4and5),the resistivity increaseswith temperature(dρ/dT>0).Based on these ob-servations,the authors argue that curves1and2correspond to an insulating phase,whereas curves4and5demonstrate a metallic behavior.At n=n3=7.25×1010cm−2(curve3),only a weak(±5%)temperature dependence was observed within theinterval of bath temperature from35mK to1K.The authors conclude that n3=7.25×1010cm−2isthe critical electron density for their sample,i.e.,itcorresponds to the metal-insulator transition point.Note that this conclusion is based entirely on thetemperature independence of curve3over a rela-tively narrow temperature range.2)Another observation,emphasized in Ref.[15],isthat in the range0.1K<T<0.4K the metal-licρ(T)-dependences(curves4,5)appear to benearly linear and thus diﬀerent from the exponen-tial temperature dependence observed in other ex-periments.KK stress that in this region the re-sistivity shows no insulating up-turn at the lowesttemperature achieved.KK consider the fact that they observed no tempera-ture dependence of the resistivity of their sample at a cer-tain concentration n=n3as a strong argument against our model[5].They point out that such a precise cancel-lation would require a special relation betweenρd(T)and the scalingβ(ρ)-function.KK argue that this relation be-tween two objects of entirely diﬀerent origin“would be a remarkable coincidence”,and is therefore unlikely. Based on the arguments listed above,authors of Ref.[15]conclude that their experiments are consistent with the existence of the zero-temperature quantum phase transition.r K HUÃ FÃFIG.1.Resistivity vs temperature,reproduced from Fig.2of Ref.[15].Electron density n in units of 1010cm −2:1- 6.85,2-7.17,3-7.25,4-7.57,5-7.85.Dashed curves A 1and A 2:ﬁts of data 1and 2by 2.29exp (403/T )1/2 and2.84exp (11.11/T )1/2,respectively (T is in mK).Solid curves B 1and B 2:ﬁts of the same data by 1.25exp (1685/T )1/3 and 2.55exp (8./T )1/3 ,re-spectively.Semi-open symbols (1′′):reﬂection of data points 1about curve 3.Curves E and F :best ﬁts of data 4and 5by ρ4(T )=1.2+1.45exp(−450/T )and ρ5(T )=0.63+1.6exp(−485/T ),respectively.Note that a rather narrow range of densities was ex-plored in Ref.[15].To show the development of the MIT-phenomenon over a wider range of densities and temper-atures,we present in Fig.2the results of Ref.[18]for a sample with a close value of µpeak .The encircled region of the {T,n }plane (region KK in this ﬁgure)indicates roughly the domain of parameters explored in Ref.[15].The temperature in Fig.2is normalized to the Fermi en-ergy E F ,in order to demonstrate that resistivities corre-sponding to quite diﬀerent Fermi energies,and therefore r s -values,exhibit similar T -dependences.A strong (50-100%)drop in ρis still present for n <∼20×1011cm −2,which corresponds to r s ∼2and ρ∼0.01h/e 2.Thus,contrary to the popular opinion,the resistance drop is not intrinsic only to the low density range (high r -values).r K H7 ()FIG.2.Resistivity vs temperature (from 0.3to ≈40K)for a wide range of densities and Fermi energies.The ﬁgure is reproduced from Ref.[18].Encircled region:domain of parameters explored in Ref.[15].Dash-dotted vertical line depicts the em-pirical temperature,T Q =0.007E F ,below which the logarithmic temperature dependence sets in.We would like to stress once again that in the region of high densities and small resistivities quantum inter-ference corrections of any kind (weak localization,ex-change and Hartree interactions,spin-orbit,etc.)would amount only to a few percent variation (of both signs)in ρwith T .The phonon contribution to the resistiv-ity of Si-MOSFETs can be shown to be negligible atleast for n <∼20×1011cm−2in the relevant T -range.Therefore,there must be some other mechanism of the T -dependence at work,neither of quantum interference nor of phonon nature.The resistance drop of non-phonon nature persists up to the highest temperature of these measurements,which is ≈35−40K for each curve in Fig.2(for conversion of densities into the Fermi-energies,use the formula:E F [K]=7.31×n [1011cm −2]).II.METAL,INSULATOR,AND CRITICAL POINT.GENERAL DISCUSSION Contrary to a common belief,a metallic phase and thus a metal-insulator transition can in principle occur in two dimensions.For example,in a hypothetical situ-ation of non-interacting electrons spin-orbit coupling re-sults in resistivity vanishing with temperature,provided that disorder is suﬃciently weak(“weak antilocalization”[3,4,19]).However,if disorder is strong enough,it leads to localization.The metallic state is stable with respect to weak and short-ranged electron-electron interactions. The Coulomb interaction between electrons for small r s overwhelms weak antilocalization and thus destroys the metallic state with zero residual resistivity.Nevertheless, the Coulomb interaction itself brings in an antilocalizing contribution to the resistivity(triplet channel).For small r s,this contribution is smaller than the one from the sin-glet channel,and thus the net eﬀect of interactions is the enhancement of localization.It may be the case that the triplet channel wins for larger r s and a metallic state be-comes possible.(In fact,this scenario follows from the RG treatment[20–22].)However,if such a state exists,it must be distinct from the conventional“disordered Fermi liquid”.So far,the experimental evidence is in contrast with the existence of a distinctly new state(see Refs.[5,7]). In fact,at higher densities the resistivity exhibits an in-sulating up-turn in agreement with the weak localization theory.We believe that in the existing experiments the true low-temperature asymptotic behavior has not yet been achieved for lower densities(in the vicinity of the transition).Therefore,one cannot interpret the crossover between dρ/dT>0and dρ/dT<0atﬁnite tempera-tures as a proof for the existence of two distinct phases. Nevertheless,it is instructive to adopt such an interpre-tation for a moment and to compare the experimentally observedρ(T)with the commonly accepted picture of a quantum phase transition.Let usﬁrst recall what is known about the transition in3D.We start our analysis by formulating the deﬁnitions of metal and insulator.Arguing about deﬁnitions is not a too rational thing to do–a deﬁnition cannot be right or wrong.We simply point out that for the well-studied case of MIT in3D,the commonly held deﬁnitions of both phases diﬀer from those adopted by KK.In3D systems,ρinsulator(T→0)→∞,whereasρmetal(T→0)isﬁnite (see[3,4,22]and references therein).In fact,dρ/dT is negative in both the metallic and insulating phases,pro-vided that the system is close to the transition point and the temperature is low enough.For non-interacting particles,the conventional sce-nario of the3D MIT is well supported by the pertur-bation theory and renormalization group arguments[ 3,4,23,22].According to this scenario,in the metallic phase close to the transitionρ(T)increases as T−αas temperature decreases.The increase ofρ(T)saturates at T<∼T sat.Upon approaching the critical point,T sat tends to zero.Therefore,exactly at the critical density theρ(T)-dependence is a power law:ρ(T)∼T−α.Inthe insulating phase,the resistivity behaves as exp(T0/T) (nearest neighbor hopping)or as exp[(T0/T)β](variablerange hopping).The exponentαis determined by mech-anisms of dephasing.As a result,it is,strictly speak-ing,non-universal.Let the dephasing rate scale with temperature as1/τϕ∝T p.Then assuming that(a) one-parameter scaling holds,(b)this parameter is the conductance,and(c)p=1,one arrives atα=1/3.Note that the electron-electron interactions can change the value ofα.Numerous experiments on3D doped semiconductors conﬁrmed the power-law behavior ofρin the critical re-gion.However,there is still no consensus regarding the value ofα:bothα=1/2[24–26]andα=1/3[27,28] have been reported.It is also possible thatαdepends on whether a semiconductor is compensated or not.(For the review of an MIT in3D see Ref.[29].)Returning to the MIT in2D,we note that the assump-tion of a temperature-independent resistivity at the crit-ical density is as doubtful as the statement thatα=1/3 in3D.Indeed,both of these points can be justiﬁed only within the one-parameter scaling.This scaling does not seem to apply universally even in the3D case.For a2D system of noninteracting electrons,this scaling predicts no metallic state at all.To get a chance to describe a metallic state,one needs to add more ingredients,e.g., electron-electron interactions,to the theory.Once we deal with a problem which is richer than localization in quenched disorder,there are no reasons to assume that the one-parameter scaling still holds(see,e.g.,Ref.[20]). Therefore,a temperature-independent critical resistivity is an assumption rather than a law of nature.As an example,consider the following density-and temperature-dependences of the resistivity(which do not follow from any of the existing theories,but do not con-tradict to any of the commonly respected general princi-ples either):ρ(n,T)=ρ0(n)+ρ1(n) T T .(1)It is natural to deﬁne the critical concentration,n c,as the concentration at which T0(n)changes sign:T0(n c)=0; for n>n c(metal),T0(n)is negative,whereas for n>n c (insulator),T0(n)is positive.It is also assumed that ρ0(n),ρ1(n)and T1(n)are smooth functions of the den-sity in the vicinity of the transition point n=n c.Ac-cording to Eq.(1),exactly at the critical point(n=n c)ρ(n c,T)=ρ0+ρ1 Tmaximum at T=−T0/α.We do not think that such a maximum is a mandatory feature of the MIT in two dimensions.)We conclude that,at least in this example, a temperature-independent resistivity is a signature of a metal rather than of a critical point.Regardless of this particular and rather artiﬁcial exam-ple,we notice that it is neither easy nor a straightforward task to determine the critical point of an MIT.Quan-tum phase transitions are zero temperature phenomena, whereas experiments are performed atﬁnite T.There-fore,to determine the critical concentration,one does not have another choice but to analyze the data taken at the lowest accessible temperatures.This analysis should prove thatρ(T)indeed acquires insulating exponential behavior as soon as the concentration gets lower than the apparent value of n c.Weﬁnd it more appropriate to use the onset of the insulating exponential behavior inρ(T)rather than the vanishing of the derivative dρ/dT for an experimental identiﬁcation of the critical point.We are not aware of any reasons to assign the meaning of the critical point to a density n∗c at which the temperature dependence of the resistivity is least pronounced,as it was done by KK.Indeed,direct measurements of the two quantities, n∗c(from the temperature independent“separatrix”)and n c(from the disappearance of the non-Ohmic hopping behavior)conﬁrm their systematic diﬀerence for high-µsamples:n∗c is larger than n c by1-5%(see,e.g.,Fig.3in Ref.[30]).A similar diﬀerence arises also from the com-parison of critical behaviors in the thermoelectric power and conductance[31].Note that four out ofﬁve electron densities in Ref.[15],n=n1−n4,fall into this ambigu-ous interval.III.EXPERIMENTAL DETERMINATION OFTHE CRITICAL POINT.We now turn to the experimental data of Ref.[15]. The authors assume that n c equals to7.25×1010cm−2 (curve3in Fig.1).According to the deﬁnition of the critical point,proposed in Sec.II,this assumption im-plies that at lower densities(n=n2=7.17×1010cm−2 and n=n1=6.85×1010cm−2),the resistivity increases exponentially as temperature decreases.It turns out that one canﬁt neitherρ(T,n=n2)norρ(T,n=n1)with a simple exponential dependence.However,ρ(T,n=n1) can be satisfactorily approximated byρexp(T,n)=ρ∗m exp[(T0m(n)/T)m](3) with either m=1/2or m=1/3(dashed line A1and solid line B1in Fig.1,respectively).The quality of theﬁt can be seen from the deviations of the experimental data from the straight line in the inset to Fig.1.Although curve2 in Fiq.1does not seem to behave exponentially,one can still try toﬁt it by function(3)in two diﬀerent ways: i)one can assume that T0m is proportional to|n−n c|.(This dependence was observed in previous studiesof the MIT in Si MOSFETs[32–34]).Then,us-ing the value of T0m for n=n1,one determines T0p for n=n2by linear extrapolation,whereasρ∗m is found byﬁtting the experimental data.The re-sults of this procedure for m=1/2are shown in Fig.1(dashed-and-dotted line C2).It is quite clear that such aﬁt does not work(the same is true for m=1/3,not shown).ii)Alternatively,one can simply make the bestﬁt of curve2by function(3)for m=1/2,1/3,treating T0m andρ∗m as independent parameters.Results of thisﬁt are depicted in Fig.1by curves A2and B2,respectively.It is customary to assume that in the vicinity of the critical point,T0m scales as |n−n c|zν.Given the values of T0m for n=n1,2, one can follow the assumption of KK that n c=n3 andﬁndzν=[ln T0m(n1)/T0m(n2)]Alternatively,one may conclude that the data of Ref.[15]do not provide evidence for a quantum phase tran-sition between two distinct states of matter,but ratherindicate a crossover between two diﬀerent types of(rela-tively)high temperature behavior.IV.RESISTIVITY OF SI MOSFETS IN THECRITICAL REGIONAs it has already been mentioned,any scenario of aquantum phase transition implies that for the densities close to the critical point n c the resistivityρ(T,n)at suﬃ-ciently high temperatures demonstrates a critical behav-ior.This means that the deviation ofρ(T,n)from theseparatrixρ(T,n c)is small for small values of the crit-ical parameter|n−n c|zν/T.Keeping this in mind,we now recall the main results of earlier experiments on the resistivity of high mobility Si MOSFETs in the criticalregion and compare them with KK’s results. Consider,for example,the resistivity of a sample with µpeak=55,000cm2/V s,reproduced from Ref.[30,37]in Fig.3.Assume for a moment that metallic and insulating behaviors can indeed be distinguished by the sign of the derivative dρ/dT,as suggested by KK.Atﬁrst glance, the application of this criterion to the region T<1K in Fig.3is rather unambiguous.Indeed,n∗c determined from the condition of T-independentρcorresponds to the dashed separatrix.The slope of the dashedline for the sample with µpeak =71,000cm 2/Vs (Fig.4a)is negative and its ab-solute value is about 10times bigger than in Fig.4b (µpeak =33,000cm 2/Vs).The reduction of the slope with µpeak persists down to relatively low mobilities un-til eventually the slope changes sign.This is seen from Fig.5a,reproduced from Ref.[37],where the authors presented cm 2/Vs.Ã7 . r K HÃr7 .FIG.5.Resistivity vs temperature for a low mobil-ity sample (Si-39).Numbers at the curve indicate thedensities in 1010cm −2.Reproduced from Ref.[37].Even for this low-mobility sample,the ρ(T )-dependence at the highest density measured (n =54.4×1011cm −2)shows no signs of saturation in the accessible temperature region (above 100mK),see Fig.5b.There is no consistent theoretical understanding of this µpeak -dependence of the slope yet.Such an understanding will probably come together with the description of the “crit-ical resistance”,which apparently changes from about 0.3e 2/h at low mobilities to several units of e 2/h at high µpeak [30,37],(see also discussion in Refs.[38,39]).On the other hand,it is natural that for the intermediatemobilities,the slope is small ,because it should vanish somewhere in the range from 5,000to 30,000cm 2/Vs.The mobility of KK’s sample is 27,000cm 2/Vs,i.e.,right in this range.Therefore one should not be too surprised by the fact that in some range of the concentrations the temperature dependence of the resistivity is rather ﬂat ,as it was observed by KK.Last but not least:a variety of the ρ(T )-dependences in the critical region indicates by itself the absence of uni-versality in observed MITs [30]and raises doubts that Si MOSFETs undergo a genuine quantum phase transition.V.“HOW COLD ARE THE ELECTRONS?”Heating of electrons by the applied source-drain ﬁeld as well as by external noise is a common problem in low-temperature transport measurements.This problem turns out to be especially serious in Si MOSFETS in the vicinity of the “metal-insulator transition”and at tem-peratures <∼100mK due to *)weak temperature depen-dence of the resistivity in this range of densities and **)weak electron-phonon coupling.As a result,the inter-pretation of resistivity measurements becomes rather am-biguous.For example,the temperature interval,in which the resistivity appears to be temperature-independent,may look to be much broader than it actually is.The goal of this section is to demonstrate the seriousness of this problem.In macroscopic Si MOSFETs,as well as in bulk Si,electrons couple to phonons only via the deformation po-tential.As this coupling is rather weak,the electron tem-perature,T e ,may exceed substantially the bath temper-ature,T bath ;it is also not easy to control T e .To provide convincing data,one has to use a “thermometer”capable of measuring T e directly.Any observable which depends on T e ,other than the (zero-magnetic-ﬁeld)resistivity it-self,can be used for this purpose.In earlier studies on high mobility Si-MOSFETs,Refs.[30,37,40–42],which reported results of the resistivity measurements down to T bath =18−25mK,the electron temperature was at-tempted to be measured viai)the amplitude of Shubnikov -de Haas (ShdH)oscilla-tions,ii)temperature dependence of the hopping resistivity,iii)linearity of I −V -characteristics,andiv)T e -dependence of the phase relaxation time.Unfortunately,results on T e in the up-to-date measure-ments of 2D MIT are neither fully consistent with each other nor convincing enough below T bath =250mK.No independent measurement of T e is reported in Ref.[15].It is only mentioned that the source-to-drain bias,U ,was less than 200µV “to ensure that the total power dissipated in the sample was less than 10−13W”.As it is not clear whether this is enough to prevent electrons from being overheated,it is worth discussing electron heating in Si MOSFETs.Electrons are driven out of the equilibrium by the ap-plied voltage and/or by external noise.Let P be the power deposited in the electron system.In the stationary state,all this power leaves the system either with elec-trons (through contacts)or with phonons.The phonon mechanism dominates at higher temperatures.If T is low enough,this mechanism can be neglected compared to electron out-diﬀusion,in which thermal conductivity of the electrons determines the heat balance.As our task is to estimate heating at T ∼100mK,we ﬁrst discuss what happens without phonons.Given the temperature increment ∆T =T e −T bath ,the power,which is carried out by electrons through the leads,can be estimated as (see Refs.[43]and also [44]for recent discussion):P =2πR,(5)where R is the resistance which diﬀers from the resistivityby the aspect ratio.To obtain a lower bound estimate of the bath temperature T h at which heating becomes strong,i.e.,∆T ≃T h ,we neglect heating due to external noise and assume that the main reason for heating is the source-to-drain bias,U .Accordingly,P =U 2/R .Ex-pressing the ratio ∆T/T bath through P via Eq.(5),we obtain∆T 2πT bath2=eU3νF LWT bath ∆Tn [K](where sis the speed of sound,and n in 1011cm −2)can be writtenas [47,48]1πΞ2m ∗k 2F a 3k F s3T bath5¯h k F s hεFaVI.TEMPERATURE-INDEPENDENTRESISTIVITYNow it is time to analyze the ﬁrst of the two main ar-guments that KK brought in favor of the “true quantum phase transition”,namely,the temperature-independent resistivity at what they believe to be the critical point.More speciﬁcally,KK claim that they observed no T -dependence (within a 10%margin)in the interval 35mK <T <1K.They analyzed the data in terms of our recent theory of Anderson localization by temperature-dependent disorder [5]and came to the conclusion that within this theory a nearly constant ρ(T )would imply a “remarkable coincidence”.We begin our discussion of this issue with summarizing brieﬂy the argument of KK.Afterwards,we present our theoretical counterarguments.The theory of Ref.[5]describes the T -dependence of the observable resistivity,ρ,in the presence of two fac-tors:i)the T -dependence of the Drude resistivity,ρd ,and ii)Anderson localization.Each of these two factors brings a T -dependence of its own and the resultant ρ(T )-dependence is described by the following scaling equation1d ln T =pβ(ρd )−1 d ln ρddTT =T max=0⇒˙y (t max )=p βd −1(15a)d 2ρβd (βd −1)[˙y (t max )]2.(15b)The third(logarithmic)derivative ofρis then found from Eq.(12)to bed3lnρβdd3yβd 3[˙y(t max)]3−3β′d8(βd−1)42β′2d−βdβ′′d (βd−1)+3β′d .(18b)(In going from(18a)to(18b),we used(15a,15b)).For a suﬃciently narrow interval of t around t max,it suﬃces to keep only the lowest(cubic)term in the Taylor expansion of lnρ(T)as a function of ln(T/T max):ln ρ(t)βd d3lnρdKK claim that the observedρ(T)stays within this mar-gin over only1.5decades,which is smaller than eventhe conservative estimate(23).(As it was discussed in Sec.V,a more realistic estimate of KK’s T-interval isone decade).A decrease inδby1%(δ=−0.01)leads to a weakly insulating behavior:ρ(T)increases by30%as T decreases by1.5decades.An increase inδby4% (δ=0.04)leads to a well-pronounced metallic behavior:ρ(T)drops by about a factor of two over1.5decades in T.These features are in quantitative agreement with theexperiment[15].(For the sake of simplicity,we assumed that parametersρ1and¯ρdo not depend on the density,i.e.,onδ.Taking these dependences into account should lead to even better agreement with the experiment.)We emphasize that noﬁne tuning between theβ-function andρd was used.If it is the competition be-tween the decrease in Drude resistivity with temperature and the localization eﬀects that leads to the maximum inthe observable resistivity,then theρ(T)-dependenceﬂat-tens out,as the density approaches the threshold for a maximum.It turns out that if the exponent p is not toobig(which is usually the case),thisﬂattening suppresses the variation ofρ(T)below the experimentally observablelevel over several decades in temperature.Even an over-simpliﬁed model[5]which does not take into account,e.g.,electron-electron interactions(except for as a pos-sible phase-breaking mechanism),can easily produce al-most constant resistivity in a temperature interval,which is two orders of magnitude wider than the experimentalone.Therefore,we do notﬁnd it too much of a“re-markable coincidence”that at some density the systemdemonstrates a fairly constantρ.We now turn to the data presented in Ref.[15].The authors emphasize strongly that at n=n3= 7.25×1011cm−2the resistivity is almost temperatureindependent in the temperature range35mK-1K.They also add thatρ(T)decreases with T for T>∼1.8K.It would be interesting to know what happens at interme-diate temperatures:1K<T<1.8K.This point beingunclear,KKﬁll in the missing temperature interval by combining their result with the one obtained on another system.Indeed,they write:“In combination with the results of Ref.[10](Ref.[52] of this paper–AMP)where the temperature-independent curve(with essentially the same value of resistivity)was observed in the temperature range250mk-1.8K in an-other2D system in silicon,we further allege that there is no observable T-dependence at n s=n c in the tem-perature range35mk-1.8K...”.(Ref.[15],p.3,second paragraph.)Parenthetically,we note that the resistivities of combined curves diﬀer by35%.As far as we understand,tempera-ture intervals of experiments on diﬀerent systems are not additive parameters,and thus the procedure described above is not justiﬁed.Summarizing this part of our discussion,we take the liberty to describe the experimental situation in the fol-lowing way:In a sample from the intermediate peak mobility range (where temperature dependence ofρis known to be quite weak at high T),one canﬁnd a density such thatρdoes not change for more than10%within about one order of magnitude in T.This statement is a re-sult of measurements on a single sample in the interval 100mK<T<1K(when a realistic estimation of the electron heating is taken into account).(It will be two samples if results of Ref.[52]for250mK<T<1.8K are taken into consideration.)In our opinion,there are neither theoretical nor exper-imental reasons to believe that the density corresponding to the weakestρ(T)-dependence is indeed the critical one.i.e.,thatρ(T→0)→∞andρ(T→0)isﬁnite for lower and higher densities,respectively.The lack of a pro-nounced temperature dependence ofρin a limited range of T does not signal any remarkable phenomenon and is well-described by a simple theory of Anderson localiza-tion in a temperature-dependent disorder(Ref.[5]).VII.ON THE APPARENT LINEAR TEMPERATURE DEPENDENCE OFρ(T)We now turn to point(2)of Ref.[15)]regarding the functional form of the observedρ(T)-dependence.Ac-cording to KK,this form is a)linear and b)qualitatively distinct from those observed by other authors.Fig.7 reproduces Fig.3b of Ref.[15],where the“almost lin-ear dependence”is demonstrated.The data in Fig.7is data5of Fig.1,displayed in truncated(T<400K)and full(T<1200mK)intervals.The truncated interval, in whichρ(T)is supposed to be“almost linear”,corre-sponds to Fig.3b of Ref.[15].We begin with an attempt toﬁt untruncated data5of Figs.1,7in the whole interval T<1.2K by an empirical expression[53]ρ(T)=ρ0+ρ1exp −T0。

Fermions at unitarity and Haldane Exclusion Statistics

a r X i v :c o n d -m a t /0606117v 2 [c o n d -m a t .o t h e r ] 29 A u g 2006Europhysics Letters PREPRINT Fermions at unitarity and Haldane exclusion statistics R.K.Bhaduri 1,2,M.V.N.Murthy 2and M.K.Srivastava 31Department of Physics and Astronomy,McMaster University,Hamilton,Canada L8S 4M12The Institute of Mathematical Sciences,Chennai 600113,India 3Institute Instrumentation Centre,Indian Institute of Technology,Roorkee 247667,India PACS.03.75.Ss –Degenerate Fermi gases.PACS.05.30.-d –Quantum statistical mechanics.Abstract.–We consider a gas of neutral fermionic atoms at ultra-low temperatures,with the attractive interaction tuned to Feshbach resonance.We calculate,the variation of the chemical potential and the energy per particle as a function of temperature by assuming the system to be an ideal gas obeying the Haldane-Wu fractional exclusion statistics.Our results for the untrapped gas compare favourably with the recently published Monte Carlo calculations of two groups.For a harmonically trapped gas,the results agree with experiment,and also with other published work.Consider a dilute gas of neutral fermionic atoms at a low temperature.In general,the low-energy properties of the gas are determined by the scattering length a ,the number density n ,and the temperature T of the gas (the eﬀective range r 0is small,so that r 0/|a |→0as a becomes large ).When the attractive interaction between the atoms is increased continuously by magnetic tuning from weak to strong,the scattering length a goes from a small negative to a small positive value.In between,there is a zero-energy two-body bound state,and |a |is inﬁnite.The gas is said to be at unitarity in this situation,and the length scale a drops out.The behaviour of the gas is expected to be universal at unitarity [1].Experimentally,if the temperature is small enough,a BCS superﬂuid is observed at the weak end,and a BEC condensate of dimers at the strong end [2].This was predicted long back by Leggett [3],who extended the BCS formalism in a novel fashion to analyse the physical situation.The BCS to BEC transition is found to be smooth,with no discontinuity in properties across the unitary point.There has been much interest amongst theorists to calculate the properties of the gas in the unitary regime (k f |a |>>1),where k f =(3π2n )1/3is the Fermi wave number of thenoninteracting gas.This is a challenging task,since there is no small expansion parameter,and a perturbative calculation cannot be done.In particular,at T =0,the energy per particle of the gas is calculated to beE5¯h 2k 2f2EUROPHYSICS LETTERS whereξ≃0.44[4].The experimental value is about0.5,but with large error bars[5].Recently, there have been two Monte Carlo(MC)ﬁnite temperature calculations[6,7]of an untrapped gas at unitarity,where various thermodynamic properties as a function of temperature have been computed.For a harmonically trapped gas,there are experimental results[8],as well as theoretical calculations[9].In the unitary regime,the thermodynamic properties have both bosonic and fermionic features[6],and it is natural to ask if in this regime the quasi-particles obey a statistics which is intermediate between the two.In this paper,we suggest on general grounds that at unitarity,so far as average properties of the system are concerned,it should behave like an ideal gas obeying the generalised exclusion statistics of Haldane[10].The deﬁnition of the statistical parameter,denoted by g(>0)in the present paper,is based on the rate at which the number of available states in a system ofﬁxed size deceases as more and more particles are added to it.The statistical parameter g assumes values0and1for bosons and fermions respectively,because the addition of one particle reduces the number of available states by g.Weﬁrst deduce the value of g for the unitary gas from theory using Eq.(1),ﬁttingξ=0.44.The value of g thus determined remains the same independent of the nature of conﬁnement as it should since the microscopic origin of the value of g depends only on the interaction between fermions and not on how the system is prepared.The application of theﬁnite temperature distribution function[11](called Haldane-Wu statistics in this paper) then enables us to calculate the temperature dependence of the energy per particle,and the chemical potential of the unitary gas.Our results for both trapped and untrapped gases are in good agreement with experiment,and MC calculations.We shall now give the rationale for using Haldane-Wu statistics at unitarity:•A strong hint that this may be the case comes from the observation that the kinetic and potential energies scale the same way when there is no length scale left from the inter-action.As is well known,Haldane-Wu statistics is realised by the Calogero-Sutherland model in one dimension[13].The potential and kinetic energy both scale the same way in this model,and both the energy densities scale as n3.Similarly,fermions in two di-mensions interacting with a zero-range potential have their kinetic and potential energy densities scale as n2,obeying Haldane-Wu statistics[14].•In the present case,a compelling evidence comes from the fact that the second virial coeﬃcient of the gas at unitarity is temperature independent[15].In exclusion statistics, the scale-invariant interaction between atoms alters the ideal Fermi(Bose)values of the (exchange)second virial coeﬃcient+(−)2−5/2by adding an interacting part[16].The above arguments are heuristic and indicative.A quantitative understanding can be obtained only when the eﬀective interaction is known fully.In the absence of such a theory,in this paper we pursue a phenomenological approach where we assume the validity of exclusion statistics on the average for quasi-particles which are otherwise non-interacting.The eﬀect of interaction is entirely subsumed in deﬁning the statistics of the quasi-particles.Weﬁrst estimate the value of the statistical parameter g from the following considerations: For Haldane-Wu statistics,the distribution function(or occupancy factor)in a single particle state with energyǫi is given by f i=(w i+g)−1,where w i obeys the relationw g i(1+w i)1−g=exp[(ǫi−µ)β],(2) whereβ=1/T,T being the temperature in units of the Boltzmann constant.Note from the above that for g=0and1,the distribution function f i reduces to the familiar bosonic andR.K.Bhaduri,M.V.N.Murthy and M.K.Srivastava :Fermions at unitarity and Haldane exclusion statistics 3fermionic forms.It is also clear that for T =0,the occupancy factor isf i (T =0)=1g2N =g 2/332M .Comparing with Eq.(1),we see that ξ=g 2/3,and choosing g =0.29gives the generally accepted value of ξ=0.44.This therefore ﬁxes the only free parameter in the model,namely,g and it should be valid independent of temperature and the nature of conﬁnement as it is the parameter which determines the statistics of quasi-particles.The main advantage of our model,however,is the calculation of the bulk properties of the gas as a function of the temperature,and this we proceed to do now.We follow the well known method (see for example the paper by Aoyama [17])for this purpose.For a given density of single-particle states D (ǫ),we haveN = ∞0D (ǫ)dǫ(w +g ).(4)For the 3−dimensional gas,D (ǫ)=C√2Nǫ−3/2f .Furthermore,ǫf =¯h 2k 2f 2T w (1+w ) ln w 1+w 0 1−g 1/2=1,(5)E 2 T w (1+w ) ln w 1+w 0 1−g 3/2.(6)In the above,w 0is the value of w at ǫ=0.For our choice of g =0.29,the Eq.(5)is solved at a given (T/ǫf )for w 0numerically,and this w 0is used in Eq.(6)next to obtain (E/Nǫf ).From the deﬁnition of w 0,it also follows that the chemical potential µat temperature T obeys the relation µǫf[g ln w 0+(1−g )ln(1+w 0)].(7)Our results for the energy per particle and the chemical potential (in units of the noninteracting Fermi energy ǫf )are plotted in Fig.1and Fig.2respectively.4EUROPHYSICS LETTERS0.2 0.40.60.811.2 1.41.61.820 0.2 0.4 0.6 0.81 1.2 1.4E /N EF T/T FFig.1–Plot of the energy per particle as a function of temperature.Both the abscissa and ordinate are in units of the free Fermi energy.The solid line corresponds to our calculations with g =0.29.The solid squares (green)with error bars are the MC calculations of [7],and the triangles (blue)are the MC calculations of [6].Our results are not sensitive to the ﬁne-tuning of the statistical parameter g .In Fig.1,we also show,for comparison,the recent MC calculated points of Bulgac et al.[6]and Burovski et al.[7].It will be seen that the agreement is very good,although the chemical potential µas calculated by us starts to diﬀer from Burovski et al result for T/ǫf >0.8).The ﬁnite temperature results are easily generalised for fermions in harmonic trap.Con-sider the fermions at T =0.The density of states D (ǫ),including a spin degeneracy factor of 2,is ǫ2/(¯h ω)3,where the oscillator parameter is deﬁned as ω=(ωx ωy ωz )1/3.It follows immediately that ˜ǫf =(3gN )1/3¯h ω,and the energy E =g 1/3(3N )4/3/4¯h ω.These results are the same as the Thomas-Fermi density functional approach of Papenbrock [18].We can easily extend these results to ﬁnite temperatures using this density of states in Eq.(4).1=3 Tw (1+w )ln w 1+w 0 1−g 2,(8)Eǫf 4× ∞w 0dw w 0 g 1+wR.K.Bhaduri,M.V.N.Murthy and M.K.Srivastava :Fermions at unitarity and Haldane exclusion statistics 5-1.2-1-0.8-0.6-0.4-0.2 00.20.40.60 0.2 0.4 0.6 0.81 1.2 1.4µ/N E F T/T FFig.2–Same as in Fig.(1)for chemical potential plotted as a function of temperature.seen that the agreement with experimental data of Kinast et al [8]as well as the many body calculation of Hu et al [9]is very good.Other thermodynamic quantities could be readily calculated.However we note that the model cannot yield the two-or many-particle correlation functions.In this regard,the situa-tion is similar to the one-dimensional Calogero-Sutherland model [12],which can be mapped on to a system of quasi-particles which obey Haldane-Wu statistics [13].But this does not help in obtaining the correlation functions,for which the full many-body calculation has to be done.Moreover,the ideal Haldane-Wu gas cannot describe super-ﬂuidity.Therefore,the main usefulness of the present approach is its ability to calculate the temperature-dependence of various bulk properties of a unitary gas with just one free parameter,namely the statistical parameter g .∗∗∗We are extremely grateful to Evgeni Burovski,Joaquin Drut and Hui Hu for sharing the results of their calculations with us.We would like to thank Akira Suzuki for useful discussions,and for invaluable help in preparing the manuscript.One of the authors (R.K.B.)beneﬁted greatly from many discussions with Duncan O’Dell.Thanks are also due to Jules Carbotte.This research was supported by NSERC of Canada.REFERENCES[1]Baker G.A.,Phys.Rev.C60,054311(1999);Heiselberg H.,Phys.Rev.A63,043606(2001);HoT.-L.,Phys.Rev.Lett.92,090402(2004).6EUROPHYSICS LETTERS0 0.511.522.50 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8E /N EF T/T FFig.3–Plot of the energy per particle as a function of temperature for the conﬁned gas.Both the abscissa and ordinate are in units of the free Fermi energy.The solid line corresponds to our calculations with g =0.29and the dashed lines corresponds to the calculations presented in Ref.[9].The experimental data is taken from Kinast et al [8].[2]Regal C.A.et al.,Nature (London)424,47(2003);Zwierlein M.W.et al.,Phys.Rev.Lett.91,250401(2003);Regal C.A.et al.,Phys.Rev.Lett.92,040403(2004);Zwierlein M.W.et al.,Nature (London)435,1046(12005);Partridge G.B.et al.,Science 311,503(2006).[3]Leggett A.J.,in Modern Trends in the Theory of Condensed Matter ,Springer-Verlag LectureNotes,Vol.115,edited by Peklaski A and Przystawa J.,(Springer-Verlag,Berlin,1980),p.13[4]Carlson J.,Chang S.-Y.,Pandharipande V.R.,and Schmidt K.E.,Phys.Rev.Lett.91,050401(2003);Perali A.,Pieri P.,and Strinati G.C.,Phys.Rev.Lett.93,100404(2004).[5]Bartenstein M.et al.,Phys.Rev.Lett.92,120401(2004);Bourdel T.et al.,Phys.Rev.Lett.93,050401(2004).[6]Bulgac A.,Drut J.E.,and Magierski P.,Phys.Rev.Lett.96,090404(2006).[7]Burovski E.,Prokof’ev N.,Svistunov B.,and Troyer M.,Phys.Rev.Lett.96,160402(2006).[8]Kinast J.,Turlapov A.,Thomas J.E.,Qijin Chen,Jelena Stajic and Levin K.,Science 3071296(2005).[9]Hu H..Xia-Ji Lu and Drummond D.,Phys.Rev.A73,023617(2006).[10]Haldane F.D.M.,Phys.Rev.Lett.67,937(1991).[11]Dasnieres de Veigy A.,and Ouvry S.,Phys.Rev.Lett.72,600(1994);Wu Y.-S.,Phys.Rev.Lett.73,922(1994);Isakov S.B.,Phys.Rev.Lett.73,2150(1994);Rajagopal A.K.,Phys.Rev.Lett.74,1048(1995).[12]Calogero F.,J.Math.Phys.10,2191(1969);10,2197(1969);Sutherland B.,J.Math.Phys.12.246(1971);12,251;Phys.Rev.A4,2019(1971).[13]Ha Z.N.C.,Phys.Rev.Lett.73,1574(1994);Isakov S.B.,Phys.Rev.Lett.73,2150(1994);Murthy M.V.N.,and Shankar R.,Phys.Rev.Lett.73,3331(1994).[14]Bhaduri R.K.,Murthy M.V.N.,and Srivastava M.K.,Phys.Rev.Lett.76,165(1996);SrivastavaR.K.Bhaduri,M.V.N.Murthy and M.K.Srivastava:Fermions at unitarity and Haldane exclusion statistics7 M.K.,Bhaduri R.K.,Law J.,and Murthy M.V.N.,Can.J.Phys.,789(2000).[15]Ho T.-L.,and Mueller E.J.,Phys.Rev.Lett.92,160404(2004).[16]Murthy M.V.N.,and Shankar R.,Phys.Rev.Lett.72,3629(1994).[17]Aoyama T.,Eur.Phys.J.B20,123(2001);cond-mat/0005336v2[18]Papenbrock T.,Phys.Rev.A72,041603(2005).。

全新版大学英语(第二版)综合教程5课后答案

UNIT 1(1) Answer: go through fire and water(2) Answer: salary(3) Answer: give(4) Answer: no peace(5) Answer: sink into(6) Answer: ambition(7) Answer: By way of(8) Answer: expressive(9) Answer: churned(10) Answer: engraved(11) Answer: not hold a candle to(12) Answer: inward24页 1) Answer: Success(2) Answer: literacy(3) Answer: significantly(4) Answer: promoting(5) Answer: appropriate(6) Answer: too(7) Answer: later(8) Answer: repetition(9) Answer: invented(10) Answer: lessUNIT 251页1) Answer: appetite(2) Answer: destructive(3) Answer: agency(4) Answer: processed(5) Answer: saturated(6) Answer: utter(7) Answer: hoisted(8) Answer: referring to(9) Answer: retrieve(10) Answer: UnfortunatelyP52 1) Peter was chasing the dog and Tom was riding the wooden horse in the garden.2) They all looked on except one young man. He took her to the hospital instantly.3) I laid charges against the company and won the case.4) If we want to stay competitive, first of all we need to modernize our factory.5) They got irrigation water from the dammed rivers.P53 1) Except in the oases the desert is almost devoid of vegetation, although somestunted, thorny shrubs grow in the western Sahara.2) The fruits growing wild in the coastal forest are edible.3) The national security agency made recommendations for improving safety standards in airplanes / to improve safety standards in airplanes.4) The Beatles enjoyed success on a scale unparalleled by any previous pop group.5) The emergence of language was a defining factor in the evolution of modern humans.1) Answer: Excluding(2) Answer: packaged(3) Answer: to boost / of boosting1) Answer: comes second to(2) Answer: infected with(3) Answer: traces of(1) Answer: vegetarian(2) Answer: are bred(3) Answer: slaughtered(4) Answer: ideal(5) Answer: reductionP56 1. sing a pop song2. died a miserable death3. live a harmonious life4. Breathing a deep breath5. dreamed a bad dream6. smiled a bitter smile7. a hard fight to fight8. sleep a troubled sleep) Answer: exclude(2) Answer: stubborn(3) Answer: devoid of(4) Answer: bow to(5) Answer: potent(6) Answer: drawbacks(7) Answer: contaminating(8) Answer: heightened(9) Answer: infected(10) Answer: come second to1) Answer: consumption(2) Answer: between(3) Answer: packed(4) Answer: evident(5) Answer: population(6) Answer: encouraging(7) Answer: grave(8) Answer: against(9) Answer: criticize(10) Answer: itself31) Answer: invitation(2) Answer: eloquent(3) Answer: concede(4) Answer: contradictory(5) Answer: conceals(6) Answer: guilty(7) Answer: generalize(8) Answer: get caught in(9) Answer: for now(10) Answer: as a last resort1) Non-smoking area. John's very intolerant of people who smoke.2) She is an interesting character, and a bit of a mystery to me.3) Because it does not reveal their marital status.4) We are planning on trekking through the Malaysian Rainforest.5) He muttered something under his breath that I couldn't understand.6) They may need to wear protective rubber gloves and clothing.7) The chairperson said sometimes unemployment tempted the youth into criminal activities.8) Though she never admitted it, the look on her face when I mentioned James' name gave her away.1) Throughout history, people have been intrigued by the question of whether there is intelligent life elsewhere in the universe.2) The hill farmers' lot has never been easy and in recent years has been assailedby a series of major crises.3) As most people in his family, Grey is a great talker when he's in the mood to talk.4) Few people find it necessary to condemn white lies on the grounds that they are not real lies.5) All the evidence of your qualifications and skills that backs up the claims you make in yourrésumé should be included.1) Answer: In general(2) Answer: that irritates(3) Answer: arrogant1) Answer: adamant(2) Answer: never indulge in(3) Answer: absurd(4) Answer: a romantic1) Answer: be aware(2) Answer: nonetheless(3) Answer: acceptable to1. was being careful with his words.2. was being polite and ate quite a bit.3. was getting impatient waiting for the waiter to come around.4. are being stupid because they run the risk of being caught and expelled from school.5. He was being a coward.6. was being a nuisance when he complained.7. I'm being serious.1) Answer: asserting(2) Answer: go along(3) Answer: because(4) Answer: part(5) Answer: Mistakes(6) Answer: exceptions(7) Answer: end(8) Answer: resort(9) Answer: dying(10) Answer: freedomUnit 3(1) Answer: convinced(2) Answer: supersede(3) Answer: personal gain(4) Answer: on the grounds that(5) Answer: devastating(6) Answer: presumptuous(7) Answer: playing God(8) Answer: manipulation(9) Answer: juggle(10) Answer: rollVocabulary (1) Answer: invitation(2) Answer: eloquent(3) Answer: concede(4) Answer: contradictory(5) Answer: conceals(6) Answer: guilty(7) Answer: generalize(8) Answer: get caught in(9) Answer: for now(10) Answer: as a last resortB 1) Non-smoking area. John's very intolerant of people who smoke.2) She is an interesting character, and a bit of a mystery to me.3) Because it does not reveal their marital status.4) We are planning on trekking through the Malaysian Rainforest.5) He muttered something under his breath that I couldn't understand.6) They may need to wear protective rubber gloves and clothing.7) The chairperson said sometimes unemployment tempted the youth into criminal activities.8) Though she never admitted it, the look on her face when I mentioned James' name gave her away.C 1) Throughout history, people have been intrigued by the question of whether there is intelligent life elsewhere in the universe.2) The hill farmers' lot has never been easy and in recent years has been assailed by a series of major crises.3) As most people in his family, Grey is a great talker when he's in the mood to talk.4) Few people find it necessary to condemn white lies on the grounds that they are not real lies.5) All the evidence of your qualifications and skills that backs up the claims you make in yourrésumé should be included.D (1) Answer: In general(2) Answer: that irritates(3) Answer: arrogant(1) Answer: adamant(2) Answer: never indulge in(3) Answer: absurd(4) Answer: a romantic(1) Answer: be aware(2) Answer: nonetheless(3) Answer: acceptable to(1) Answer: be aware(2) Answer: nonetheless(3) Answer: acceptable toUsage 1. was being careful with his words.2. was being polite and ate quite a bit.3. was getting impatient waiting for the waiter to come around.4. are being stupid because they run the risk of being caught and expelled from school.5. He was being a coward.6. was being a nuisance when he complained.7. I'm being serious.Comprehensive (1) Answer: go along(2) Answer: honesty(3) Answer: straightforward(4) Answer: indulge in(5) Answer: What about(6) Answer: dodge(7) Answer: assert(8) Answer: absurd(9) Answer: resort(10) Answer: juggle(1) Answer: asserting(2) Answer: go along(3) Answer: because(4) Answer: part(5) Answer: Mistakes(6) Answer: exceptions(7) Answer: end(8) Answer: resort(9) Answer: dying(10) Answer: freedomUnit 42. Narration.3. In chronological order.4. For example, the odd animals stored in the upper apartment of the laboratory; his lunch at school; his observation of the whole group of haemulons, etc. Details such as these are omitted because they are of little help in bringing out the theme of the essay —how the professor's teaching method enabled him to see the importance of close observation and thus exerted a life-long influence on his academic career.(1) Answer: after lingering(2) Answer: resuscitate(3) Answer: sloppy(4) Answer: gaze(5) Answer: loathsome(6) Answer: ghastly(7) Answer: sideways(8) Answer: in despair(9) Answer: concluded(10) Answer: with infiniteVocabulary (1) Answer: for myself(2) Answer: concluded(3) Answer: infinite(4) Answer: internal(5) Answer: misery(6) Answer: mode(7) Answer: ventured(8) Answer: visible(9) Answer: observation(10) Answer: commendedB 1) I lay in bed feeling thoroughly wretched.2) It is fragrant with the smell of apple blossom.3) They are fine specimens of the veteran revolutionaries.4) I'd like to enroll in the modern art course if it is not too late.5) The taste is slightly bitter, and it has a strange odor.C 1. The scheme does nothing to help families on low incomes and is sure to provoke / call forth / draw / arouse criticism.2. Jenny is terribly uncertain as to whether Bob is the right boy for her.3. These goods bear no resemblance to those I saw printed in the advertisements.4. In China, where black hair and black eyes are the norm, her blond hair and blue eyes are rather conspicuous.5. We did not have time for a rehearsal before the performance because of the delay of our flight.D (1) Answer: an energetic(2) Answer: ardent(3) Answer: The investigation(4) Answer: a wretched(5) Answer: To leave them to their own devices(1) Answer: entrusted(2) Answer: explicit(3) Answer: did not commend itself to(4) Answer: look him in the face(5) Answer: turning to(1) Answer: my subsequent(2) Answer: The latter(3) Answer: part with(4) Answer: with reluctance(1) Answer: Enrolling(2) Answer: specimen(3) Answer: leave him to his own devices(4) Answer: investigation(5) Answer: By and by(6) Answer: content with(7) Answer: entrusted(8) Answer: reluctance(9) Answer: infectious(10) Answer: observation(11) Answer: mode(12) Answer: grounded in(1) Answer: known(2) Answer: only(3) Answer: doing(4) Answer: assistance(5) Answer: assignment(6) Answer: simply(7) Answer: But(8) Answer: turned(9) Answer: singled(10) Answer: because54. Yes, the essay ends well by asserting the number of skeptics on global warming is growing smaller, thus highlighting the point that global warming has undoubtedly become a reality. In a way, the ending echoes the beginning, helping render the whole piece more coherent.(1) Answer: densely populated(2) Answer: uninhabitable(3) Answer: thrown into(4) Answer: migrate(5) Answer: contaminate(6) Answer: respiratory(7) Answer: widen the range(8) Answer: incidence(9) Answer: adjusting(10) Answer: wildlife15% in the last four years.Answer: percentagesuch as New York, London, Los Angeles, Tokyo, creating a continuous, 24-hour Earth Day celebration.Answer: zone3) The cloud absorbs sunlight, heating the stratosphere (平流层) up but stoppingAnswer: warmthAnswer: diverseNorth and South.Answer: widen6) It's such a beautiful city, it's a shame we didn't have more timeAnswer: look aroundthey can only dimly perceive.Answer: in the face ofAnswer: in perspective9) Most aquarium (水族馆) plants come from tropical and subtropical areas, with aAnswer: temperatethis chapter needs to be supplemented by practical experience.Answer: theoretical1)Its profits shrank from $5 million to $1.25 million in the last global financialcrisis.2) They will have to adhere to the cultural norms of the organization in orderto be successful with their database project.3) My hometown is/lies halfway in between Salk Lake City and Denver.4) I saw waves battering (against) the rocks at the bottom of the cliff.5) Flood waters washed away the only bridge connecting the village to the outsideworld.2)1) Your report on the new car park is fine, but why don't you beef it up withsome figures?2) There is a wide variation among Internet providers in cost, features, software,reliability and customer service.3) Poverty is one of the reasons for the high incidence of crime in thisneighborhood.4) I suggested we sing and dance for the elderly people in the nursing home,and all my roommates were in favor of my idea.5) Doctors who are compelled to work 36 hours at a stretch cannot possibly befully efficient.1) Much of the loss of biodiversity currently being experiencedpopulations wiping out entire ecosystems for development and single crop farming.habitat. We already of deforestation on global warming, but do we stop to think about the thousands of animal and insect species that(die off, attribute, diverse, scary)(1) Answer: is attributed to(2) Answer: diverse(3) Answer: the scary(4) Answer: are dying off2) In August 2005, some organizations predicted that a temperature increase of 2 °C above the pre-industrial level couldfor sea levels and biodiversity. At the current level of climate change, this prediction could become a reality in 10-15years. (trigger, esteemed, overwhelming)(1) Answer: from esteemed(2) Answer: trigger(3) Answer: overwhelming3) With huge amounts of carbon dioxide and greenhouse gases in the atmosphere global(magnify, go up, turmoil, throw into, melt)(1) Answer: melting(2) Answer: throwing(3) Answer: into turmoil(4) Answer: be magnified(5) Answer: going up(1) Answer: contaminated(2) Answer: contaminate(3) Answer: contamination(4) Answer: uncontaminated(5) Answer: habitable(6) Answer: habitation(7) Answer: inhabit(8) Answer: uninhabited(9) Answer: uninhabitable(10) Answer: inhabited Comprehensive(1) Answer: beef up(2) Answer: coastal(3) Answer: in favour of(4) Answer: residents(5) Answer: theoretical(6) Answer: disastrous(7) Answer: battered(8) Answer: shrinking(9) Answer: migrate(10) Answer: washed away(11) Answer: Scary(12) Answer: humanity(1) Answer: predicting(2) Answer: accuracy(3) Answer: basis(4) Answer: collide(5) Answer: atmosphere(6) Answer: melts(7) Answer: affected(8) Answer: actions(9) Answer: striving(10) Answer: technologies Unit 64. To contrast differences, the author uses:And that... is where the contrast between...... on the other hand...So Grant and Lee were in complete contrast...To transit from contrasting differences to comparing similarities, the author uses:Yet it was not all contrast, after all.To compare similarities, the author uses:Each man had...Daring and resourcefulness they had, too...Lastly...(1) Answer: Different(2) Answer: underlying aspiration(3) Answer: in common(4) Answer: much alike(5) Answer: to begin with(6) Answer: acute(7) Answer: handicaps(8) Answer: indomitable(9) Answer: refusal(10) Answer: on his feet1) As a man who has reached his present who has experienced the tough side of life, he knows more about life than most people.Answer: the hard way2) Bob made to Ann before their marriage that he would break his gambling addiction for good.Answer: solemn3) upon list of all the things she needed, from padlocks to picture frames.Answer: wrote outrespect.Answer: champion5) The President has announced a policy for the nation's wetlands, which calls forAnswer: ownershipvisiting their families at the weekend.Answer: privilegeshe didn't know very much about romance, just enough to know that it seemed to be a force that did not like to be tamed.Answer: To be sureas anyone else in the school.Answer: handicap9) His plan met with resistance from her family, who regarded the idea as wholly unrealistic.Answer: surge(1) Answer: cut the ground from under(2) Answer: feetB 1) The committee aims to achieve reconciliation between the two opposing parties.2) The management's refusal to increase the minimum monthly grant.3) Public places such as metro stations, theaters, and museums have been made more accessible to the disabled.4) His love for the countryside brought forth a series of remarkable watercolors.5) It embodies the ideals of freedom and equality.C 1) The books are keyed to the interests of children.2) We will not stand by and let the small village schools get closed for lack of funds.3) Their response was in effect a refusal to our request though they didn't turn it down explicitly.4) Generous to a fault, he paid for all the expenses.5) We shall always feel we are deeply in your debt.Dspeed. Asked about reasons for the great success, he said,company: cooperative with regard to making decisions, and trustingfellow workers. Every employee has a strong sensethe company and boundless the work." (underlying, prosper, enthusiasm, obligation, to begin with, in relation to)(1) Answer: prospered(2) Answer: the underlying(3) Answer: To begin with(4) Answer: in relation to(5) Answer: of obligation to(6) Answer: enthusiasm for2) Most chapters of this book are dedicated to the effects of games on children.According to the author, in performing and observing actions, likecooperate with each other in the competition. (personality, implicit, collision)(1) Answer: the collision(2) Answer: implicit(3) Answer: personality3) Martin Luther King and Rosa Parker had a They were black peopleand inequality and to try to gain control over Faced with greatall men are created equal. (destiny, in common, combat, notion, hang on)(1) Answer: in common(2) Answer: to combat(3) Answer: own destinies(4) Answer: hung on(5) Answer: the notionE 1) Both of these follow their dreams and finally became successful lawyers.Answer: set out2) Yesterday a lone his attempt to beat the record for crossing the Atlantic.Answer: set off3) Fortunately the wound was treated before infection (感染Answer: set inyour purse.Answer: set aside5) A new middle school, where excellent teaching is given by friendly and helpfulAnswer: set upbackground color for floor and walls first.Answer: set aboutAnswer: set offhis stock more than his customers, often refusing to sell a cherished piece to a disliked buyer.Answer: set upbattle.Answer: set up10) Rome her neighbors by her ancient fame and lasting influence. Answer: set apart1. According to the manager, what he wants is a simple yet effective sales plan.2. Usually he was a serious man, yet this joke reduced him to hearty laughter / set him laughing heartily.3. The governor has put forward a series of policies to cut the state budget, but the effects of the new measures have yet to be seen.4. He is not yet 20, but his technical control, confidence, brilliance and intellectual depth display an outstanding maturity.5. John wouldn't let me see his essay, because he hasn't finished it yet.6. Maybe the reason scientists have yet to receive signals from extraterrestrialintelligence is that there isn't any extraterrestrial intelligence sending signals.7. She knew the sensible thing to do was to leave the place as soon as possible, yet she wanted to stay.8. Her selective yet comprehensive exhibition draws mainly from public collections, among them many of the United States' most distinguished libraries.(1) Answer: personalities(2) Answer: embody(3) Answer: underlying(4) Answer: collision(5) Answer: leadership(6) Answer: ownership(7) Answer: ideals(8) Answer: champion(9) Answer: the hard way(10) Answer: prospered1) Answer: indication(2) Answer: sensitive(3) Answer: career(4) Answer: resign(5) Answer: supply(6) Answer: disciplined(7) Answer: promoted(8) Answer: criticized(9) Answer: surrender(10) Answer: respected。

STREGA WP1M1 mirror substrates - LIGOStrega wp1m1镜基板- LIGO

C an tilever d isp lacem en t an g le (rad s)
Amplitude ring-down of uncoated silicon cantilever for the third bending mode at f~1.9 kHz at 120K
0.04
cantilever displacement angle exponential fit 0.02
0.00
-0.02
-0.04 0
200
400
600
800
Time (s)
Coated silicon cantilever
• Results – third bending mode at ~1.9kHz
(b)
1x10 -5
m e c h a n i c a l l ofs(sw )
(a)
1x10 -6
Coated silicotion
• Interpretation: calculating the energy stored in the coating layer ctd
We can approximate the energy stored in the compressed half of the beam to be equal to the energy stored in the stretched half. Scaling by a factor of two therefore gives the ratio of energies stored in the surface layer (coating) and the bulk to be:
• Experiments suggest

化学动力学三四章习题和答案

《化学反应动力学》第三、四章习题1、试推导A + B → P ，当其为二级反应r = k ［Ａ］［Ｂ］时，其积分速率方程为： ln{1 + Δ0（P 0 - P ∞）/ [A]0（P t – P ∞）} = ln [B]0 / [A]0 +Δ0k t式中，P 为用物理仪器测定的体系的某种物理性质（如吸光系数），该性质与浓度有线性关系，Δ0 = [B]0 - [A]0解：设B 过量A +B → Pt = 0 [A]0 [B]0 P 0 = [A]0λA + [B]0λB + λM (1) t = t [A]0-x [B]0-x P t = ([A]0-x) λA + ([B]0-x) λB + λM (2) t = ∞ [B]0 - [A]0 P ∞ = ([B]0- [A]0) λB + λM (3) 据二级反应速率方程：)]([][)]([][ln ][][1000000x B A x A B B A kt ---= x x A A B kt B A --+=-000000]B [][ln ][][ln)][]([ kt A B x A x B 00000][][ln ][][ln ∆+=-- （1）-（3）得：P 0 – P ∞= [A]0(λA +λB ) , 即：00A ][A P P B ∞-=+λλ （1）-（2）得：P 0 – P t =λA x + λB x由上面两式可得：000][A P P P P x t ∞--= 0000000000][][][][ln ][][ln A P P P P A A P P P P B x A x B t t ∞∞------=-- 000000000000][][][][][][][][ln A P A P P A P A P A P A P B P B t t +--+--=∞∞ )(][][][][][)][]([ln00000000∞∞∞∞-+-+--=P P A P A P A P A P B P A B t t)(][)(][))(][]([ln 00000∞∞∞--+--=P P A P P A P P A B t t ))(][)(1ln(000∞∞--∆+=P P A P P t 故： kt A B P P A P P t 000000][][ln ))(][)(1ln(∆+=--∆+∞∞ 2、反应Np 3+ + Fe 3+ → Np 4+ + Fe 2+。