GRB jet beaming angle statistics

扇形束重建图像的一簇新算法
潘晓川
【期刊名称】《《CT理论与应用研究》》
【年(卷),期】1999(008)002
【摘要】本文提出重建扇形束（fan－beam）图像的一簇新的混合算法。

这簇新
方法使得重建扇形束图像既不产生偏差，又可改进扇形束图像中的噪声性质。

同时，通过理论分析和计算机数字模拟，证明此新方法不仅运算速度快欲常用方法，并且具有比常用的扇形束渺滤波反投影的算法更为优越的数学和统计特性。

【总页数】4页(P44-47)
【作者】潘晓川
【作者单位】美国芝加哥大学放射学系
【正文语种】中文
【中图分类】R814.42
【相关文献】
1.等角扇形束投影的CT重建算法研究 [J], 敖波;曾亚斌;吴伟;金信鸿;祝忆春
2.扇形束CT超短扫描优质重建算法研究 [J], 马建华;颜刚;陈凌剑;陈武凡
3.由相叠的低分辨扇形束投影数据重建高分辨CT图像 [J], 张朋;郭明焕
4.基于小波的扇形束CT图像局部重建算法 [J], 郭荣;乔志伟
5.关于扇形束超短扫描几何下图像重建算法的改进研究(英文) [J], 叶丽颖;王金平因版权原因，仅展示原文概要，查看原文内容请购买。

a rXiv:as tr o-ph/972182v121Fe b1997The Halo Beaming Model for Gamma-Ray Bursts R.C.Duncan 1&Hui Li 2ReceivedABSTRACTWe consider a model for gamma-ray bursts(GRBs)from high-velocity neutron stars in the galactic halo.In this model,bursters are born in the galactic disk with large recoil velocities V r,and GRBs are beamed to within emission cones of half-angleφb centered on V r.We describe scenarios for magnetically-channeled GRBs that have such beaming characteristics.We then make detailed comparisons of this halo beaming model(HBM)to data from the3rd BATSE Catalog and from the Pioneer Venus Orbiter experiment,for both GRB intensity and angular position distributions.Acceptablefits to observations of over1000bursts are obtained forφb=15◦−30◦and for a BATSE sampling depth of D∼180kpc,which corresponds to a peak burst luminosity of∼1040ergs s−1.Present data favor a truly isotropic(cosmological) model over the HBM,but not by a statistically compelling margin(<∼2σ).The HBM makes the distinctive prediction that the galactocentric quadrupole moment cos2Θ −1/3for bright,nearby GRBs is large,even though the dipole moment cosΘ remains near zero.Bursters born in nearby external galaxies,such as M31,are almost entirely undetectable in the HBM because of misdirected beaming.We analyze several refinements of the basic HBM: gamma-ray intensities that vary with angle from the beam axis;non-standard-candle GRB luminosity functions;and models including a subset of bursters that do not escape from the galaxy.We also discuss the energy budgets for the bursters,the origins of their recoils,and the physics of burst beaming and alignment.One possible physical model is based on the magnetar model of soft gamma repeaters(SGRs).Empirical bounds on the rate of formation and peculiar velocities of SGRs imply that there exist∼104to∼107aged SGRs in the galactic halo within a distance of100kpc.The HBM gives an acceptablefit to observations only if it satisfies some special conditions(φb≈20◦,uniform bursting rate)which are possible,but for which there are no clear and compelling theoretical justifications.The cosmological burst hypothesis is more generic and thus more attractive in this sense.Subject headings:gamma rays:bursts—stars:magneticfields—stars:neutron —galaxy:halo1.IntroductionThe Burst and Transient Source Experiment(BATSE)on the NASA Compton Gamma-Ray Observatory has revealed a nearly isotropic but inhomogeneous sky distribution of gamma-ray bursts(Meegan et al.1992;Meegan et al.1996).An additional major constraint on GRB theories,revealed by the Pioneer Venus Orbiter(PVO)experiment,is that the intensity distribution of bright bursts is consistent with a locally-uniform density of sources in Euclidean space(Fenimore et al.1993).These facts are simply and generically accounted for if the faintest observed GRBs come from cosmological distances(e.g.,Paczy´n ski1995). An alternative possibility,that we will focus on here,is that the bursts come from the extended halo or corona of our galaxy.Galactic halo models for GRBs werefirst considered and tested against data by Fishman et al.(1978)and Jennings&White(1980).Reasons for favoring halo models have been summarized by Lamb(1995).In galactic halo GRB models,it is often supposed that the bursters are neutron stars, since such compact stars are conjectured to be capable of producing intensefluxes of hard, non-thermal photons;while their small size could drive burst variability on submillisecond time scales.Other reasons for favoring neutron stars include observations of spectral lines, which are at present controversial,and a possible connection of classic GRBs with the March5,1979gamma ray burster(e.g.,Duncan&Thompson1992,hereafter DT92)which was localized to an angular position lying within a young supernova remnant(Cline et al.1982;Rothschild,Kulkarni&Lingenfelter1994).The displacement of this burster from the center of the supernova remnant indicates that it acquired a velocity∼1000km s−1at birth(DT92).Neutron stars with such velocities will escape the galactic disk,and move into the halo.Recent analysis of the1979March5th event lends support to its association with classical GRBs(Fenimore,Klebesadel,&Laros1996,hereafter FKL).In addition, many high-velocity radio pulsars(V>500km s−1)have been observed(Lyne&Lorimer1994;Frail1996).For these reasons,we will focus on theories of GRBs from high-velocity neutron stars(HVNSs)born in the galactic disk,asfirst suggested by Shklovskii&Mitrofanov (1985).In particular,we will consider the“halo beaming model”(HBM)proposed earlier (Duncan,Li,&Thompson1993,hereafter DLT;Li,Duncan,&Thompson1994,LDT;Li &Duncan,1996a,1996b;Bulik&Lamb1996).Alternative models for GRBs from HVNSs in the galactic halo invoke a delayed onset of bursting activity at a time∼107years after birth in the disk(Li&Dermer1992,hereafter LD92),which has been applied in several different physical contexts(Colgate&Leonard1994;Lamb,Bulik&Coppi1996;Woosley &Herant1996);and weakly-bound bursters orbiting in a nonspherical galactic potential (Podsiadlowski,Rees&Ruderman1995,hereafter PRR).Models for bursters born in the galactic halo(Eichler&Silk1992;Hartmann1992;Salpeter&Wasserman1993;Wasserman &Salpeter1994),or the Magellanic Clouds(Fabian&Posiadlowski1993),are beyond the scope of this paper.In§2,we will describe our basic model for the galactic population of bursters,and briefly review its physical motivations.In§3we present our basic Monte Carlo model results.We discuss how these results arise,and what they could imply for halo GRB theories.We also explore the model’s sensitivity to different beaming angles.In§4we make detailed quantitative comparisons of the model with BATSE and PVO data.In particular, we study moments of the angular position distribution in bright subsets of observed GRBs, which is potentially a sensitive model discriminant.In§5we estimate the GRB repetition rate and the energy requirements for bursters.We also discuss two candidate power sources that could satisfy these requirements:magnetic energy and accretion energy.Severalrefinements of the basic HBM are investigated in§6,namely:gamma-ray intensities that vary with angular position within the beam,GRB luminosity functions,and models witha significant subset of bursters on bound orbits in the galactic halo.In§7we give our conclusions and outline future observational tests.Note that we consider possible physical mechanisms for halo GRBs,involving neutron stars with unusually strong magneticfields,in§2.3,§2.4,§5,§6.3,and Appendix B.These sections could be read separately from the rest of the paper,since they might apply in a non-beamed model context(cf.§7.2).2.The Halo Beaming Model:Physical Motivations2.1.Model AssumptionsWe will assume that GRBs are emitted by HVNSs,which emanate from the galactic disk like a“wind”extending into the galactic corona(Shklovskii&Mitrofanov1985). The HBM does not require that the bursters or their peculiar velocities have any favored orientation in a galactic coordinate frame,but it does invoke the physically-plausible condition that gamma-rays are produced only within a cone of angular radiusφb about the star’s magnetic axis±−→µ.The magnetic axis is furthermore assumed to be roughly aligned (within∼20◦)with the stellar recoil velocity V r.We discuss the physics of such beaming and alignment in§2.2.The particular version of HBM which we will analyze quantitatively in§3and§4below has the following simple properties:[1]bursters are born at positions distributed like young Pop.I stars in the galactic disk;[2]with randomly directed recoils V r=1000km s−1;[3] they emit GRBs at a constant rate,with[4]constant luminosity,and[5]the gamma ray emission is beamed parallel and anti-parallel to V r,within an angular radiusφb that we will vary.GRB beaming in a direction correlated with V r makes the observable burst distributioncomply with the BATSE dipole isotropy and the PVO brightness–distribution constraints, for the following reasons(see Figure1).Since the HVNSs are freely–streaming out of the galaxy,their mean density diminishes with distance from the galactic center as n∼r−2. However,the fraction of escaping bursters that are potentially detectable at Earth increases in proportion to the transverse area of their beaming cones,∼r2.These two trends cancell, making the effective number of bursters increase linearly with the sampled volume(i.e.,an apparent“constant density”of detectable stars)within a“core radius”R c∼R o/φb,where R o∼8.5kpc is a galactic disk dimension.In the HBM,this produces the the observed “homogeneous”distribution of bright GRBs found by the PVO experiment.Furthermore, since most bursters are undetectable when they are at small r,the observable dipole anisotropy of GRB positions in the direction of the galactic center is greatly reduced.At distances larger than R c,all bursters are detectable at Earth(or nearly all;see exceptions discussed in§3),and the n∝r−2free-streaming fall-offof burster density prevails, accounting for the“boundedness”( V/V max <0.5)found by BATSE.A more detailed illustration of the geometrical effect of burst beaming is given in Figure2.To understand thisfigure,consider for a moment an idealized model in which all stars are born precisely at the galactic center(GC),and move out in random directions on straight–line trajectories,each emitting GRBs into a cone of half-opening angleφb around its velocity vector.Bursters lying outside the circles in Figure2,or within the lens-shaped intersection of circles between the GC and Earth,are then the only ones that can be detected at Earth.There is a large“zone of avoidance”(ZOA),within which all bursters are invisible.Figure2actually shows only the2-D cross-section of this ZOA.The true ZOA is the volume of revolution of the pictured shape about the line between Earth and the galactic center.The boundary in Figure2at which stars just become observable(edge of the ZOA)is part of a circle with radius D sun/sinφb,where D sun=8.5kpc is the distance from Sun to the galactic center.In the realistic situation,bursters are born throughout the galactic disk,with a peak birth rate at∼4–5kpc from the center,where the greatest concentration of Population I stars are located(van der Kruit1987).Each birthplace in the disk then has its own ZOA, scaling up or down in linear size with the distance between the birthplace and Earth.One must add the weighted distributions of detectable bursts together to get the total observable GRB distribution.There is no simple,analytic way to do this,so in what follows we will use Monte Carlo methods.We will also calculate realistic trajectories in the galactic potential, rather than assuming straight lines.2.2.Physics of Beaming and AlignmentGamma-ray bursts from high-B environments tend to be beamed alongfield lines because of the transverse pair-creation opacity(e.g.,Riffert,M´e sz´a ros&Bagoly1989; Ho,Epstein&Fenimore1990)and because gamma rays produced by such mechanisms as curvature radiation and Compton upscattering are strongly beamed alongfield lines (e.g.,Sturrock1986;Dermer1990and references therein).Even locally–isotropic emissions are strongly beamed when they occur in a relativistic outflow channeled by a magneticfield(e.g.,Yi1993).Gamma emissions induced by sheared Alfv´e n waves in a neutron star magnetosphere can also be highly beamed(Melia&Fatuzzo1991;Fatuzzo&Melia 1993).Beaming obviates theγ–γopacity limit for GRBs(Krolik&Pier1991;Baring1993; Fenimore Epstein&Ho1993).The HBM invokes the additional(assumed)property that burster recoil velocities areapproximately aligned with magnetic dipole—and hence burst beaming—axi:−→V r −→µ.This requires that the rotation axis −→Ωis also roughly aligned with−→µ,at least to within theburst opening angleφb.Is −→V r −→µa plausible assumption?Any recoil mechanism which imparts impulses to the stellar surface with a coherence time longer than the rotation period of the star willtend to make −→V r ±−→Ωbecause of “rotational averaging.”One such recoil mechanism isanisotropic neutrino emission during the first ∼10s after formation (§2.4below).If itis also true that −→µ ±−→Ω(at least to within the beaming angle φb )then the alignmentcondition of the HBM is satisfied.Such near–alignment between −→µand ±−→Ωis expected if the neutron star magnetic fieldis generated by large-scale dynamo action,as in familiar stellar and planetary dynamos.The “magnetar”model,described below (§2.3),is one scenario involving such a dynamo.Vacuum magnetic torques increase alignment (Michel &Goldwire 1970;Davis &Goldstein 1970)after the star spins down past the “death line”at which magnetospheric currents are quenched,at age t D ∼105(B dipole /1014G)−1yrs (Chen &Ruderman 1993).Before this point is reached,currents might exert counter-aligning torques of comparable strength (e.g.,McKinnon 1993and references therein).Note that the GRB beaming angle φb is the sum of the intrinsic beaming angle due to gamma-ray opacity and radiative effects,and the r.m.s.scatter in the angle between −→Ωand −→µ.Gamma production mechanisms such as curvature radiation and Compton upscattering could operate at significant altitudes in the magnetosphere,where field linesare tilted with respect to −→µ,yielding larger values of φb than one would estimate under theassumption of near-polar gamma emission (as adopted by Riffert,M´e sz´a ros &Bagoly 1989;Ho,Epstein &Fenimore 1990).Values as large as φb ∼20◦are possible.2.3.Magnetically-powered Bursts?The HBM might apply in a variety of different physical contexts.Here we will briefly discuss the“magnetar”model for GRBs(DT92).The magnetar idea has previously been invoked in models of soft gamma repeater bursts(Thompson&Duncan1995,hereafter TD95)as well as in models of continuous X-ray emissions from SGRs and from anomalous X-ray pulsars like1E2259+586(Thompson&Duncan1996,TD96).Magnetars are hypothetical neutron stars which are born with dipolefields in excess of B Q≡me2c3/e¯h=4.4×1013G.How might such stars form?Neutron stars are hot and convective during thefirst∼10seconds after they form.They also undergo strong differential rotation(TD93).Large-scale dynamo action might operate efficiently during this time for neutron stars with mean rotation periods below a threshold comparable to the convective overturn time,as predicted byα–Ωdynamo theory.There is evidence for such a threshold in magnetically-active main-sequence stars(e.g.,Simon1990).This threshold effect could give rise to a bimodal population of neutron stars(“pulsars and magnetars”) with a factor>∼102difference infield strength(DT92).A more detailed explanation of this magnetar formation hypothesis is given by Duncan&Thompson(1996,DT96).Seven distinct estimates of thefield in the bursting neutron star SGR0526−66seem to indicate B>1014G(TD95).3Magnetars spin down too rapidly to be observed as radio pulsars.DT92conjectured that GRBs areflare-like reconnection events in a galactic halo population of such strongly-magnetized neutron stars.We will show in§4.5that the available magnetic energy is sufficient to power observed GRBs,in the context of the HBM(see also PRR).Magnetic reconnection is a theoretically advantageous energy source for GRBs,for several reasons.Gravitational and nuclear energy releases generally occur in bulk baryonic matter,which has many degrees of freedom into which energy can thermalized and degraded via adiabatic expansion(e.g.,Piran&Shemi1993).Flares in a neutron star magnetosphere, on the other hand,can be“clean,”exciting only photon and pair degrees of freedom to a first rge scale electromotive forces induced by reconnection will generally accelerate pairs,and may produce hard,non-thermal spectra via Compton upscattering, synchrotron emission,and curvature radiation(Sturrock1986).Indeed,GRBs have many qualitative similarities to stellarflares.Both are transient energy releases,with chaotic variability over a wide range of time scales;and both have spectrally-hard non-thermal components(e.g.,Murphy et al.1993;Ramaty&Mandzhavidze1993).The similarities are so strong that a minor fraction of bursts in the BATSE catalog might actually be intense events from nearby,non-neutron,flare stars(Liang&Li1993).Stochastic avalanche models give a goodfit to GRB time profiles(Stern&Svensson1996)and to many properties of solarflares(Lu et al.1993).If the V r–B dipole correlation observed in some samples of radiopulsars(e.g.,Cordes 1986;Stollman&van den Heuvel1986)is applicable for all neutron stars,then the mean magnetar recoil velocity would be>∼10times larger than the mean for radiopulsars(but see Lorimer,Lyne and Anderson1995).Several mechanisms predict unusually large recoils for magnetars,sufficient to propel them into the galactic halo,as explained in§2.4below.Because of the dynamo origins of magnetarfields,one expects that initially −→Ω,±−→V rand±−→µare roughly aligned in magnetars.4As the star spins down,magnetospheric currents might drive some degree of misalignment;however,once the star spins down past the“death line”at time∼105(B dipole/1014G)−1yrs(Chen&Ruderman1993),further spindown certainly enforces alignment(Michel&Goldwire1970;Davis&Goldstein1970), since the magnetostatic stellar distortion in magnetars is large enough to damp nutations of −→Ωabout−→µ(Goldreich1970,DLT).Incidentally,this is probably not true in old,spun-down(P rot>4s)radiopulsars.We conclude that −→V r is likely to be roughly aligned with±−→µinold magnetars,where±−→µis also the axis of beamed gamma emissions(DLT).5 This scenario for classic GRBs assumes that magnetars retain strong dipole magnetic firge-scale magnetic instabilities that reduce B dipole(Flowers&Ruderman1979) might be suppressed by toroidalfield components in the stellar interior,as expected for fields generated viaα–Ωdynamos,or perhaps by other mechanisms(TD93§14.2).Note that a dipolefield anchored in the stably–stratified liquid interior of a magnetar cannot be greatly distorted by spindown-induced crustal tectonic drift(Ruderman1991)because magnetic stresses dominate tensile stresses in the crust.This might differ markedly from the circumstance in ordinary radiopulsars(Ruderman1991).The interactions of vortexlines in the neutron superfluid with superconductingflux tubes are also probably irrelevant for young magnetars,because the interiorfield is probably strong enough to suppress superconductivity.A young magnetar’sfield evolves predominantly via ambipolar diffusion in the interior and Hall fracturing in the crust(TD96).As the interiorfield decays, superconductivity will appear in the mantle and perhaps the core;but this probably happens only after rapid spindown has driven most superfluid vortex lines out of the interior.Further discussion of magnetar evolution is given by PRR.2.4.Neutrino Magnetic RecoilsA dipole anisotropy of only∼0.03in the neutrino emission from a young,hot neutron star would impart a recoil velocity∼1000km s−1to the star(Chugai1984).Here we briefly review mechanisms whereby neutrino anisotropies could be magnetically induced. Purely hydrodynamic(non-magnetic)mechanisms for producing neutron star recoils have been proposed by Janka&M¨u ller(1994),Shimizu,Yamada&Sato(1994),Burrows& Hayes(1996),and in references quoted therein.A strong magneticfield affects neutrino emissions from the beta processes n→p+e−νand p+e−→n¯ν,as calculated by Dorofeev,Rodionov&Ternov(1985),and from neutrino scattering processes as calculated by Vilenkin(1995).Dorofeev et al.and Vilenkin furthermore estimated the neutron star recoils resulting from these processes,in the uniform-field idealization.This is a macroscopic manifestion of parity-nonconservation in the weak interactions.6In a realistically non-uniform magneticfield,the back-reaction of magnetic stresses on convective energyflow,along with the neutrino opacity variations outlined above,would give rise to“neutrino starspots,”analogous to sunspots,and thus produce neutron star recoils of a magnitude estimated in DT92and§13of TD93.These calculations are based on standard Weinberg-Salam weak interaction theory. If neutrinos have mass,on the other hand,a strong magneticfield will shift resonantflavor-changing(MSW)oscillations,thereby also inducing an anisotropy in the emergent neutrinoflux from a nascent neutron star(Kusenko&Segr`e1996).All of these mechanisms produce recoils which vary directly—usually linearly—with the mean magneticfield strength:V r∝B.Thus if any of these mechanisms dominate, one would expect much larger recoils for magnetars than for pulsars.Several non-neutrino recoil mechanisms which also might operate more efficiently in magnetars than in pulsars were discussed by DT92.7To estimate neutrino magnetic recoils realistically,one must know the strength,coherence length and coherence time of the magneticfield during the epoch of maximum neutrino luminosity.This is not possible at present.However,if the magneticfield approaches equipartition with the free energy of differential rotation and/or the convective fluid mixing—which is strongly in the MHD limit—then B>∼1016G at the neutrinosphere (TD93).Such strongfields could be present when most of the neutrino energy is radiated away,even if the surface dipolefield,which is frozen-in tens of seconds later by the onset of stable stratification in the liquid interior(Goldreich&Reisenegger1992),is smaller by a factor of∼10or∼30.Recoils V r>∼103km s−1are plausible.3.Galactic Halo Model ResultsBy integrating a large number(∼106)of neutron star trajectories in a realistic galactic potential,we have derived the sky distribution of HVNSs.In the model described here(§3 and§4),all stars have|V r|=1000km s−1;thus they move in nearly straight lines and eventually escape from the galaxy(but see§6.3).We used a Monte Carlo code developed by Li&Dermer(1992),which in turn follows many of the prescriptions of Paczy´n ski(1990)and Hartmann,Epstein&Woosley(1990). In particular,we adopt van der Kruit’s(1987)model for the spatial distribution of young Pop.I stars which spawn neutron stars in the galactic disk.The trajectories of∼106 neutron stars were numerically integrated in a realistic galactic potential(Miyamoto& Nagai1975)8including a dark halo cutoffat radius R H=70kpc.The escape velocity fromthe center of this model potential is V esc∼600km s−1.This increases only logarithmically with R H,reaching800km s−1for R H=200kpc,thus our results are probably insensitive to R H over its plausible range(LD92).Deviations from sphericity in the dark halo were neglected(but see PRR).We also do not include the potential of M31since stars withV r∼103km s−1are negligibly perturbed by M31within the BATSE sampling depth we found of<200kpc(§4.1).In Figure3we show HBM numerical results for several angular statistics,plotted as functions of sampling depth D about the Earth.That is,thefigure shows cumulative angular statistics for all detectable bursters at distances from Earth that are less than or equal to the value of D on the horizontal axis.The topmost plot of Figure3shows the galactocentric dipole moment cosΘ ,whereΘis the angle between a burst and the galactic center.The second plot shows the disk-like quadrupole, sin2b −1/3,where b is galactic latitude; sin2b <1/3implies that sources are concentrated toward the disk.The third plot shows the galactocentric quadrupole, cos2Θ −1/3.The bottom plot of Figure3 shows V/V max ,a statistic which is related to the slope of the cumulative log N—log P brightness distribution,as explained below.Within each subplot of Figure3,the various lines correspond to different beaming anglesφb as described in thefigure caption.We have cut offtheφb=10◦and5◦curves for D≤10kpc because our Monte Carlo sampling statistics at smaller D are too poor.For example,the number of detectable stars at D≤3kpc in our Monte Carlo model is only ∼55forφb=5◦,whereas it is∼104for in the unbeamed case(φb=90◦).Increasing the sampling depth D(moving to the right in Figure3)is tantamount to including fainter and fainter bursts.Eventually the BATSE sampling depth is reached;atthis point,if the model is tofit observations,all the plotted statistics must simultaneously match BATSE values to within observational uncertainty.The most recent published BATSE results9are plotted in Figure3at a value D=180kpc.Before discussing these results,we must explain the significance of V/V max .The V/V max statistic was invented for the study of the quasars(Schmidt1968),and wasfirst applied to GRBs by Schmidt,Higdon&Hueter(1988).For scintillation counter experiments, V/V max is the average of(C/C min)−3/2over all bursts,where C is the peak counts(in a set time interval)and C min is the threshold for detection.C min can vary with background noise and other effects(e.g.,“overwrites”);however,for a source population that is uniformly distributed in static Euclidean space, V/V max is equal to0.5 regardless of how the threshold varies;and V/V max <0.5indicates that the density of bursters diminishes with distance,for standard-candle sources.Because C min is variable, the V/V max statistic is useful in the study of inhomogeneous data sets only when trying to answer the yes/no question:“Is the observed brightness distribution consistent witha uniform density of sources distributed in Euclidean space?”(e.g.,Band1992;Petrosian 1993).In Figure3,we use V/V max in a purely illustrative way.Our model values of V/V max are ideal values that would be found by an instrument with a uniform detection threshold, i.e.,no variations in the noise or the threshold settings.Since the C min values in the BATSE catalog are not highly variable(Meegan et al.1996)preliminary comparisons with BATSE,as in the bottom panel of Figure3,will not lead us astray.However,the most accurate way to compare burst observations with theory is tofit the observed distribution of peak photonfluxes to Monte Carlo models which have been realisticallyfiltered for detection incompleteness(e.g.,Lubin&Wijers1993).This is what we do in our actual statistical comparisons of the HBM with the BATSE catalog(§4).Note that the model curves for angular statistics in Figure3implicitly assume a detector with uniform sky coverage;i.e.,we have idealized that the detector is equally capable of detecting bursts from any location on the celestial sphere.We have corrected for the imperfect sky coverage of BATSE by shifting the data points in Figure3appropriately (see footnote9).In§4we will take the opposite approach:using the raw BATSE data and filtering the Monte Carlo model to take into account BATSE’s imperfect sky coverage.What can be learned from Figure3?Beaming evidently increases V/V max to nearly 0.5for nearby bursters,D∼30kpc.This can make the model satisfy PVO constraints,as we show quantitatively below.Beaming also evidently reduces cosΘ .Theφb=20◦case fits BATSE observations over an appreciable range of sampling depths D>100kpc.Note that the largest deviations from isotropy in Figure3occur for bright(i.e., small-D)subsets of the observable bursts.A distinctive signature of the HBM is thatcos2Θ −1/3is positive for bright bursts.This can be understood as follows.The(bright) bursters which become visible at Earth before reaching the remote galactic halo are the oneswhich happen to be born with beaming axis−→µ(and hence,with −→V r)pointed approximatelytoward or away from Earth(to within∼φb).Because most bursters are born within the Solar circle in the galactic disk,the brightest ones tend to be seen in that direction and toward the galactic anticenter,if they have already moved past the Earth on their way out of the galaxy.Thus cos2Θ >1/3for bright bursts,while the dipole moment cosΘ remains small.。

雷达信噪比方程嘿，朋友们！今天咱们来聊聊雷达信噪比方程这个有点神秘又超有趣的东西。

你可以把雷达想象成一个超级侦探，在黑暗中寻找小坏蛋（目标）呢。

这时候，信噪比方程就像是侦探的秘密武器。

信噪比（SNR）等于发射功率（Pt）乘以目标的雷达散射截面积（σ）乘以接收天线增益（Gr）乘以发射天线增益（Gt），再除以（4π）的平方乘以距离（R）的四次方乘以系统损耗（L）乘以玻尔兹曼常数（k）乘以带宽（B）乘以接收系统噪声温度（Tn）。

哇塞，这方程看起来就像一串神秘咒语，不过别怕，咱慢慢解读。

发射功率（Pt）呢，这就像是侦探的大嗓门。

功率越大，就好像侦探的声音越洪亮，能传播得更远去寻找目标。

要是发射功率小得可怜，那就像侦探在那细声细语，目标都听不到它的呼喊，肯定不行啦。

目标的雷达散射截面积（σ）就好比目标的“显眼程度”。

比如说，一个巨大的金属物体，就像在黑暗中穿着一身闪闪发光的大袍子，雷达散射截面积就大，很容易被发现。

而一个小小的、表面还吸波的东西，就像一个擅长隐身的小忍者，很难被探测到。

接收天线增益（Gr）和发射天线增益（Gt）呢，这就像是侦探的超级耳朵和大喇叭。

增益越大，耳朵就越灵敏，喇叭就越响亮，信号的收发就越顺畅。

再说说距离（R）的四次方在分母上。

这距离啊，就像一个超级贪婪的大怪兽。

距离稍微增加一点，对信噪比的影响就像被大怪兽狠狠地咬了一口，信号衰减得特别厉害。

就好像你和朋友说话，离得远一点声音就小很多，离得越远越听不清。

系统损耗（L）就像是路上的小坑洼。

每一个小坑洼都会削弱信号的强度，要是坑洼太多，信号就像一个磕磕绊绊的小可怜，到达目的地的时候都快散架了。

玻尔兹曼常数（k）、带宽（B）和接收系统噪声温度（Tn）这几个参数，就像是捣蛋鬼。

它们会产生噪声，就像一群小老鼠在那吱吱叫，干扰我们的信号。

噪声温度越高，就像小老鼠越兴奋，叫得越欢，越影响我们的信噪比。

你看，这个雷达信噪比方程虽然复杂，但把每个部分想象成这样有趣的东西，是不是就好理解多啦？它就像一个精心编排的舞蹈，每个参数都有自己的角色，缺了谁或者谁没做好，这个舞蹈就不完美了，雷达这个超级侦探也就不能很好地完成任务啦。

固体火箭尾焰雷达散射截面数值计算引言：固体火箭的尾焰是由燃烧产生的高温高压气体流所形成的，由于尾焰的特殊性质，它对雷达信号的散射截面产生一定的影响。

雷达散射截面（RCS）描述了雷达波向目标散射的能量，是一个重要的参数，用于评估目标的探测和追踪性能。

本文将介绍固体火箭尾焰雷达散射截面的数值计算方法。

1.固体火箭尾焰的特性2.固体火箭尾焰雷达散射截面的计算方法2.1几何光学法几何光学法是最简单直观的计算方法，它假设尾焰是具有一定形状的简单几何体，并计算其表面的反射和散射。

这种方法适用于简单形状的尾焰，但对于复杂形状的尾焰效果较差。

2.2多散射法多散射法是一种复杂的计算方法，它考虑了尾焰内部的多次反射和散射。

该方法通过数值计算求解尾焰内部的电磁场分布，再根据散射机制计算出雷达散射截面。

这种方法需要大量的计算，但可以得到较为准确的结果。

2.3光学理论法光学理论法基于电磁波的传播和反射原理，通过计算电磁波在尾焰中的传播和散射来计算雷达散射截面。

这种方法的优点是计算简单，适用于较为复杂的尾焰形状。

3.实验测量方法实验测量方法是通过实验手段直接测量固体火箭尾焰的雷达散射截面。

常用的实验方法包括雷达测量法、扫描测量法和探测火箭轨道法。

实验测量法具有较高的精度和准确性，但需要考虑实验环境和其他因素的影响。

4.固体火箭尾焰雷达散射截面数值计算的挑战与展望固体火箭尾焰雷达散射截面的数值计算面临着一些挑战，如尾焰形状的复杂性、尾焰内部的多次反射和散射效应等。

未来的研究可以结合理论计算和实验测量方法，开展更深入的研究，以提高固体火箭尾焰雷达散射截面的数值计算精度和准确性。

结论：固体火箭尾焰雷达散射截面的数值计算是一个复杂而重要的问题。

通过几何光学法、多散射法和光学理论法的计算，以及实验测量方法的应用，可以对固体火箭尾焰的雷达散射截面进行较为准确的评估和预测。

随着科学技术的不断发展，我们可以进一步完善计算方法，提高固体火箭尾焰雷达散射截面数值计算的精度和准确性，在军事和航天领域中发挥更大的作用。

Polym.Bull.(2013)70:105–115DOI10.1007/s00289-012-0784-0O R I G I N A L P A P E RGrazing-incidence wide-angle X-ray diffraction studyon molecular aggregation state of imprinted polyimidefilm before and after hard bakingSudu Siqing•Hui Wu•Hiroki Yamaguchi•Takamichi Shinohara•Osami Sakata•Atsushi TakaharaReceived:12February2012/Revised:15May2012/Accepted:30May2012/Published online:12June2012ÓSpringer-Verlag2012Abstract Nanoimprint lithography(NIL)was carried out on precursor of polyimide (PI),poly(amic acid)film,and then hard baking to obtain imprinted PIfilm.The molecular aggregation states of imprinted PIfilms before and after hard baking were investigated by grazing-incidence wide-angle X-ray diffraction comparing with the one offlat PIfilm.It was found that NIL and hard baking can strongly affect the molecular aggregation states of PIfilm.Before hard baking,PI chain is aligned parallel to the line direction on the line.After hard baking,the alignment in ordered domain was changed to that the PI molecule of which chain axis is perpendicular to the line direction is significantly increased,while,PI molecule of which chain axis is parallel to the line direction is decreased after hard baking.Through comparing with theflat PI, crystallinity of imprinted PIfilm has been significantly enhanced.Keywords:GI-WAXDÁMolecular aggregation stateÁImprinted PIfilmÁHard bakingIntroductionPolymer crystallization is notoriously difficult to control.Very recently,many studies have shown that nanoimprinting can control the molecular orientation of S.SiqingÁH.YamaguchiÁT.ShinoharaÁA.Takahara(&)Graduate School of Engineering,Kyushu University,744Motooka,Nishi-ku,Fukuoka819-0395, Japane-mail:takahara@cstf.kyushu-u.ac.jpH.WuÁA.TakaharaInstitute for Materials Chemistry and Engineering,Kyushu University,Kyushu,JapanO.SakataJapan Synchrotron Radiation Research Institute,Mikazuki,Sayo,Hyogo679-5198,Japanpolymers such as poly(9,9-dioctylfluorene-co-benzothiadiazole)[1],poly(vinyli-denefluoride)[2]and poly-3(hexyl thiophene)[3].Polyimides(PIs)are well-known high-performance engineering plastics exhibiting outstanding properties[4–7]. Especially,aromatic PIs are widely used as liquid crystal(LC)alignment layers[5–7].Nowadays,the molecular aggregation state of aromatic PIs has been addressed in many reports[8–13].For instance,in1993,grazing-incidence X-ray scattering (GIXS)measurements of the near surface structure of an aromatic PI was reported [8].In1995,near-surface molecular aggregation states of rubbed aromatic PIfilms were studied by GIXS[9].However,the molecular aggregation sate of nanoim-printed PIfilm has not been reported.The main reason is that it is difficult to directly fabricate nanoimprinting on PIfilm due to its high glass transition temperature(T g,350–550°C).On the other hand,recently,there are a few reports mentioned about fabrication of nanopattern on precursor of PI then hard baking at a certain temperature[14,15].However,it has not been revealed whether the hard baking after patterning will effect on the molecular aggregation state.In this study,nanoimprint lithography(NIL)was used to fabricate line pattern on precursor of PI then hard baking to obtain imprinted PIfilm.The molecular aggregation states of imprinted PIfilms before and after hard baking were studied by grazing-incidence wide-angle X-ray diffraction(GI-WAXD)comparing with the one offlat PIfilm.ExperimentalPreparation of PAA solutionOxydianiline(ODA)and pyromellitic dianhydride(PMDA)were chosen as reactants.N-methyl-2-pyrrolidone(NMP)99.0%was used as a solvent.PMDA, ODA,and NMP were purchased from Wako Chemicals Co.,Osaka,Japan,without any further purification before use.The PAA solution was prepared by solution condensation polymerization at ambient temperature and at a composition of 14.5wt%by solvent.Fabrication of imprinted line pattern on PAAfilmPAA solution which was diluted to a composition of2.9wt%solids(Viscosity is 0.011Pa s.)by NMP solvent was spin coated on silicon(Si)wafer(2,500rpm and 90s).The Si wafers were cleaned in a fresh piranha solution(a mixture of98% H2SO4and30%H2O2with a volume ratio of7:3)at90°C for6h,rinsed thoroughly with copious amounts of deionized water and dried with nitrogen gas. Soft baking of PAAfilm was carried out on a hot plate at80°C for3min.NIL was executed by a commercial nanoimprinter(NM-0401,Meisyo Kiko Co.,Ltd.).The imprint process was carried out in a vacuum condition to avoid air bubbles. The imprint pressure was applied at20MPa from80to160°C.Table1shows the nanoimprinting process of imprinted PIfilm.The fabrication process is shown in Fig.1.To decrease the adhesion between the polymer and the mold,Si moldTable1Nanoimprinting process of imprinted PIfilmStep1Step2Step3Step4Step5Pressure(MPa)03030300 Temperature(°C)80801603030 Holding time(s)1060601010(Kyodo International Inc.Japan)was treated with Optool DSX(Daikin Industries Ltd.,Japan;0.1wt%in Methoxy-nonafluorobutane[HFE7100])for1min,rinsed by HFE7100,and dried under vacuum.Hard baking processHard baking was carried out in a muffle furnace(EYELA,KDF S-70).The temperature was ramped from room temperature to300°C in1h and held at 300°C for1h,and then cooling down naturally.After hard baking,the thickness of PIfilms was ca.200nm,which was tested by AFM scratch method. Preparation offlat PIfilmTheflat PI was prepared by the same concentration of PAA solution(2.9wt%),spin coating condition,thermal annealing process(soft-baking process and hard baking process).Measurement of GI-WAXDGI-WAXD was used to get the information about the structures by in-plane measurement and out-of-plane measurement[16–18].The schematic geometries are shown in Fig.2.The measurements were performed with a BL13XU beamline at the Japan Synchrotron Radiation Research Institute(SPring-8)using an incident X-ray with wavelengths k of0.100nm.The scattering vector,q,in specular reflectivity is defined by q=(4p/k)sin h,where k and h are the wavelength and detected angle of the X-ray beam,respectively.The data collection time was1s pera step,and the angular interval was0.58.The critical angle,a c,is calculated to be0.1168.The measurement at a i=0.088and0.168indicates the information of the surface and the bulk aggregation states,respectively.Results and discussionTopography of imprinted PI film before and after hard bakingAFM images of imprinted PI film before hard baking (imprinted PI bef )and imprinted PI film after hard baking (imprinted PI aft )were shown in Fig.3a,b.Line-heights of imprinted PI bef and imprinted PI aft are ca.115±10nm.In imprinted PI bef film,the average of pitch-width (line-width plus groove-width)is ca.1,595nm.After hard baking,the average of pitch-width has changed into ca.1,541nm.The result shows the shrinkage of the film after hard baking.The line-height of nanopattern slightly increased by hard baking might be induced by the significant shrinkage of the residual layer.In addition,it is found the groove become wider because of the shrinkage of the line.2Diffracted Lattice Plane Incident X-ray 2f Incidentq xy800100012001400160018002000115 nm1595 nm n m Before HardBaking5 µm 800100012001400160018002000120 nm 1541 nm n m After Hard Baking 5 µm 157.46147.2400Molecular aggregation state of flat PI filmIt is known that PI films do not exhibit definitive crystalline diffraction peaks,which indicates the absence of large domains with 3-D positional order.Hence,the ordered domains with mesomorphic order between crystalline and amorphous phase in the film can be interpreted as liquid–crystalline-like (LC-like)ordered ually,the molecular aggregation state of PI film is identified as a mixture of a LC-like ordered domain and an amorphous matrix [19].The schematic is shown in Fig.4.A PI chain is depicted as a strip with a rectangular cross-section owing to the planar chain conformation with benzene rings [19].In Fig.4,the abbreviation ‘‘ch-pack’’represents the packing structure of PI chains in LC-like ordered domain [19].The GI-WAXD profiles of flat PI film are shown in Fig.5.From in-plane profiles,no any peak can be observed in the surface region and bulk,indicating that the molecular aggregation state is disordered in the direction of the film plane.Out-of-plane profiles of flat PI film exhibit a peak at q =13.0nm -1(d =0.48nm)and a peak at 19.0nm -1(d =0.33nm).The former peak at q =13.0nm -1can be DianhydridePIDiamineCh-packIn-plane Ch-pack Out-of-plane Amorphous Liquid-crystalline-like ordered domainFig.4Schematic illustration of molecular aggregation state of PI film which is a mixture of a liquid-crystalline-like ordered domain and an amorphous matrixFig.5a GI-WAXD profile of flat PI film.b Schematic of molecular aggregation state for flat PI filmcharacterized as the PI chain packing (ch-pack)[8,12].This peak is partly overlapped by an amorphous halo.Another peak at q =19.0nm -1(d =0.33nm)is indexed as p –p stacking (abbreviated as p -stack)[19].In out-of-plane profiles,the appearance of ch-pack and p -stack indicates there is the intermolecular ordering of PI chain packing (a and b axes),which are perpendicular to the PI chains [19],in the direction of normal to the substrate.In the previous report,it has been revealed that aromatic PI chains,which are oriented in the film plane,are more densely packed in the direction of film thickness rather than in the film plane.So our results agree with the previous reports [19,20].According to intensity of peaks,in the out-of-plane profiles,the ordering in bulk is more than that in the surface region.It might result from the tensile stress due to the mismatch of thermal expansion coefficient between the PI films and the substrates,which induces PI molecules being close to the substrate is more ordered than that being close to the surface.Molecular aggregation state of imprinted PI film before hard bakingUsually,two azimuthal angles of incident X-ray were utilized to estimate molecular orientation [3,9].The 08direction of incident X-ray is parallel to the nanoimprinting line direction (for short:line direction),while,the 908direction of incident X-ray is perpendicular to the line direction.The schematic is shown in Fig.6.As for in-plane measurement,when the 08incident X-ray was applied,the diffracted lattice plane is almost parallel to the line direction and normal to the substrate.In the same way,when the 908incident X-ray was applied,the diffracted lattice plane is almost perpendicular to the line direction and normal to the substrate.In case of out-of-plane,the GI-WAXD profiles should be same when we use these two directions for the measurement of the crystalline and amorphous structure.As for LC-like structure,the profiles are different because the packing of PI chains is relatively disordered in the LC-like ordered structure due to variation in conformation around the chain axis and the distribution of interchain distances (Fig.4)[19].Figure 7a,b shows the GI-WAXD profiles of imprinted PI bef film obtained by in-plane measurement and out-of-plane measurement,respectively.In Fig.7a,there is no any peak appear in the 08direction GI-WAXD profiles in the surface region and in bulk.It illustrates the molecular aggregation state is disordered at the directionOut-of-plane 0˚IncidentX-ray Diffracted X-ray 90˚ IncidentX-ray DiffractedX-rayd -spacing In-plane0˚IncidentX-ray 90˚ IncidentX-ray d -spacing DiffractedX-ray d -spacing Diffracted X-ray d -spacing Fig.6Schematic illustration for diffracted lattice plane of line patterned film at 08and 908direction of incident X-raywhich is perpendicular to the line direction at the film plane.In the 908direction GI-WAXD profiles,there is a prominent peak at ca.4.0nm -1(d =1.57nm)in the surface region and in bulk.This peak can be indexed as a (002)peak which represents the periodic structure along the molecular chains (c axis)[8].Since (002)peak appeared in 908direction GI-WAXD profiles,it illustrates the molecular chain axis included in the ordered domain is aligned along the line direction.The absence of (00l)peak in 08direction GI-WAXD profiles confirms the molecular chain axis is aligned along the line direction.The reason of this orientation is that,when the viscous polymer melt flow into the mold cavities by the imprinting pressure,the polymer molecules were aligned along the flow direction during the thermal embossing [3,21,22].In addition,the intensity of (002)peak in the surface region is twice higher than that in bulk.This indicates the molecular chain parallel to the line direction in surface region is more ordered than that in bulk.On the other hand,a peak indexed as ch-pack (a and b axes)can be observed in bulk rather than in surface region in 908direction GI-WAXD profiles.It might be due to the influence of the residual layer of the nanoimprinted structure [3].During NIL,the partly imidization of film lead to the shrinkage of the film.The shrinkage of the lines made the residual layer stretched perpendicular to the line direction,which might cause that the ch-pack (a and b axes)appears in the 908direction GI-WAXD profiles of residuallayer.Fig.7a In-plane GI-WAXD profiles of imprinted PI bef film.b Out-of-plane GI-WAXD profiles of imprinted PI bef film.c Schematic of molecular aggregation state for imprinted PI bef filmFigure7b shows the out-of-plane GI-WAXD profiles of imprinted PI beffilm.In 08direction GI-WAXD profiles,the profile in the surface region overlapped with the one in bulk.Both show a peak indexed as p-stack which represents the ordering which is perpendicular to chain axis.It illustrates the aggregation state is same in the surface region and in bulk,which shows slight ordering of perpendicular to PI chain. On the other hand,in908direction GI-WAXD profiles in bulk,there are a peak of (101)at q=10.2nm-1,a peak of ch-pack at q=13nm-1and a peak of p-stack (q=19.5nm-1).It illustrates there is the intermolecular ordering of PI chain packing(a and b axes).This ordering can be regarded as resultant primarily from the residual layer.Furthermore,the absence of(00l)peaks in the out-of-plane profiles indicates that the PI chains included in the ordered domains are preferentially oriented parallel to thefilm plane.On the basis of these results,the aggregation state in imprinted PI beffilm,the schematic is shown in Fig.7c,can be interpreted as a mixture of LC-like ordered domains and an amorphous matrix,in which long-range3-D positional order is absent.In addition,molecular chains are aligned along the line direction on the line.Molecular aggregation state of imprinted PIfilm after hard bakingFigure8a shows GI-WAXD profiles of imprinted PI aftfilm obtained by in-plane measurement.Four peaks appear:(002),ch-pack,(010),and p-stack.The peak of (002)represents the periodic structure along the molecular chains(c axis).The appearance of(002)peak in in-plane GI-WAXD profiles illustrates the molecular aggregation state of imprinted PI aftfilm has changed to be more ordered than the one of theflat PIfilm due to NIL.It is because that the stretching during imidization for theflatfilm is non-directional,and that,for the imprinted PIfilm,the stretching during imidization is perpendicular to the line direction.At08direction GI-WAXD profiles,(002)peak represents molecular chain axis perpendicular to the line direction.On the contrary,at908direction GI-WAXD profiles,it represents molecular chain axis parallel to the line direction.According to the intensity of (002)peak of imprinted PI aftfilm and imprinted PI beffilm at08direction and at908 direction,PI molecule of which chain axis is perpendicular to the line direction is significantly increased,while,PI molecule of which chain axis is parallel to the line direction is decreased after hard baking.To explain this phenomenon,the change of topography after hard baking should be considered.The phenomenon might result from the shrinkage of thefilm during hard baking.Since there is an adhesion between the substrate and thefilm,thefilm is stretched on substrate when the shrinkage occurs.Meanwhile,due to the existence of line pattern,the shrinkage of each line induced the direction of stretching is toward the direction which is perpendicular to the line direction.Therefore,the shrinkage of thefilm with line pattern and the stretching which is perpendicular to the line direction might be main reason of the alignment change.The schematic of shrinkage and stretching of the film with line pattern during hard baking is shown in Fig.8d.In in-plane GI-WAXD profiles of imprinted PI aftfilm,the(010)peak appears at 08direction,but the absence at908direction.In addition,the intensity of(002) peaks at08direction are higher than that at908direction.According to thisobservation,it is confirmed the ordering at the direction which is perpendicular to the line direction is more than the one at the direction which is parallel to the line direction.It is also confirmed the film was stretched at the direction which is perpendicular to the line direction during hard baking.It is worth to notice that the ch-packs in in-plane GI-WAXD profiles of imprinted PI aft film are much stronger than those of imprinted PI bef film.Because the ch-pack representing the mesomorphic order region can be interpreted as LC-like ordered domains.The ch-pack changed to be stronger indicates the intermolecular ordering degree of PI chain has been more in the film plane after hard baking.Out-of-plane GI-WAXD profiles of imprinted PI aft film are shown in Fig.8b.The peak at ca.q =13.0nm -1is PI chain packing.Another peak is p -stack.TheFig.8a In-plane GI-WAXD profiles of imprinted PI aft film.The side figure shows magnified views of the (002)peaks.b Out-of-plane GI-WAXD profiles of imprinted PI aft film.c Schematic of molecular aggregation state for imprinted PI aft film.The alignment became random after hard baking and PI molecule of which chain axis is perpendicular to the line direction is significantly increased.d Schematic of shrinkage and stretching of the film with line pattern during hard bakingabsence of(00l)peaks in the out-of-plane profiles indicates that the PI chains included in the ordered domains are preferentially oriented parallel to thefilm plane. Molecular aggregation state in the surface region and in bulk is different at908 direction owning to the effect of the residual layer.Compared theflat PI with imprinted PI aftfilm,obviously,line-patterned NIL has a strong influence on the molecular aggregation state parallel to thefilm plane (obtained from in-plane profiles)than the one perpendicular to the substrate (obtained from out-of-plane profiles).Theflat PI shows the molecular aggregation state of perpendicular to the substrate is more order,however,imprinted PI aftfilm presents molecular aggregation state is more order at parallel to thefilm plane because of the existence of line pattern.Meanwhile,NIL made PI crystallinity has significantly enhanced.According to the appearance of(010)peak and stronger ch-pack,the ordering domain of imprinted PI aftfilm approaches‘‘crystalline-like’’[8]. ConclusionsIn summary,GI-WAXD was used to analyze the molecular aggregation state of imprinted PIfilm before and after hard baking comparing with the one offlat PI.It is found that PI chain included in the ordered domains are preferentially oriented parallel to thefilm plane in three kind of PIfilms.In imprinted PI beffilm,PI chain is aligned parallel to the line direction on the line.After hard baking,the alignment of molecular aggregation state of imprinted PI beffilm became random owning to the shrinkage and the stretching of thefilm during hard baking.Owing to the same reason,in the ordered domains,PI molecule of which chain axis is perpendicular to the line direction is significantly increased,while,PI molecule of which chain axis is parallel to the line direction is decreased after hard pared with theflat PI,crystallinity of imprinted PI aftfilm has been significantly enhanced,which approaches‘‘crystalline-like’’.Through this study,NIL can be a novel method to enhance crystallinity of PIfilm.Acknowledgments The synchrotron radiation X-ray diffraction experiments were performed at the BL13XU in the SPring-8with the approval of JASRI(Proposal No.2011A1485).This study was supported by a Grant-in-Aid for the Global COE Program‘‘Science for Future Molecular Systems’’from the Ministry of Education,Culture,Science,Sports and Technology of Japan.References1.Friend RH,Zheng ZJ,Yim KH,Saifullah MSM,Welland ME,Kim JS,Huck WTS(2007)Uniaxialalignment of liquid-crystalline conjugated polymers by nanoconfinement.Nano Lett7:987–992 2.Jonas AM et al(2005)Nanoscale control of polymer crystallization by nanoimprint lithography.Nano Lett5:1738–17433.Hu WC,Aryal M,Trivedi K(2009)Nano-confinement induced chain alignment in ordered p3htnanostructures defined by nanoimprint lithography.ACS Nano3:3085–30904.Sroog CE(1976)Polyimides.J Polym Sci Macromol Rev D11:161–2085.Ree M(2006)High performance polyimides for applications in microelectronics andflat paneldisplays.Macromol Res14:1–33Polym.Bull.(2013)70:105–1151156.Seo DS,Kobayashi S,Nishikawa M(1992)Study of the pretilt angle for5CB on rubbed polyimidefilms containing trifluoromethyl moiety and analysis of the Surface atomic concentration off/c(percent)with an electron spectroscope for chemical-analysis.Appl Phys Lett61:2392–23947.Vanaerle N,Barmentlo M,Hollering RWJ(1993)Effect of rubbing on the molecular-orientationwithin polyimide orienting layers of liquid crystal displays.J Appl Phys74:3111–31208.Factor BJ,Russell TP,Toney MF(1993)Grazing incidence x-ray scattering studies of thinfilms ofan aromatic polyimide.Macromolecules26:2847–28599.Toney MF,Russell TP,Logan JA,Kikuchi H,Sands JM,Kumar SK(1995)Near-surface alignmentof polymers in rubbedfilms.Nature374:709–71110.Factor BJ,Russell TP,Toney MF(1991)Surface-induced ordering of an aromatic polyimide.PhysRev Lett66:118111.Ando S,Takizawa K,Wakita J,Azami S(2011)Relationship between molecular aggregationstructures and optical properties of polyimidefilms analyzed by synchrotron wide-angle X-ray dif-fraction,infrared absorption,and UV/Visible absorption spectroscopy at very high pressure.Mac-romolecules44:349–35912.Ando S,Takizawa K,Wakita J,Kakiage M,Masunaga H(2010)Molecular aggregation structures ofpolyimidefilms at very high pressure analyzed by Synchrotron wide-angle X-ray diffraction.Mac-romolecules43:2115–211713.Stohr J,Samant MG,Luning J,Callegari AC,Chaudhari P,Doyle JP,Lacey JA,Lien SA,Pu-rushothaman S,Speidell JL(2001)Liquid crystal alignment on carbonaceous surfaces with orien-tational order.Science292:2299–230214.Chiou DR,Yeh KY,Chen LJ(2006)Adjustable pretilt angle of nematic4-n-pentyl-40-cyanobiphenylon self-assembled monolayers formed from organosilanes on square-wave grating silica surfaces.Appl Phys Lett88:13312315.Chiou DR,Chen LJ,Lee CD(2006)Pretilt angle of liquid crystals and liquid-crystal alignment onmicrogrooved polyimide surfaces fabricated by soft embossing ngmuir22:9403–940816.Yakabe H,Tanaka K,Nagamura T,Sasaki S,Sakata O,Takahara A,Kajiyama T(2005)Grazingincidence X-ray diffraction study on surface crystal structure of polyethylene thinfilms.Polym Bull53:213–22217.Honda K,Yakabe H,Koga T,Sasaki S,Sakata O,Otsuka H,Takahara A(2005)Molecularaggregation structure of poly(fluoroalkyl acrylate)thinfilms evaluated by synchrotron-sourcedgrazing-incidence X-ray diffraction.Chem Lett34:1024–102518.Yamaguchi H,Kikuchi M,Kobayashi M,Ogawa H,Masunaga H,Sakata O,Takahara A(2012)Influence of molecular weight dispersity of poly{2-(perfluorooctyl)ethyl acrylate}brushes on theirmolecular aggregation states and wetting behavior.Macromolecules.45:1509–151619.Wakita J,Jin S,Shin TJ,Ree M,Ando S(2010)Analysis of molecular aggregation structures of fullyaromatic and semialiphatic polyimidefilms with synchrotron grazing incidence wide-angle X-rayscattering.Macromolecules43:1930–194120.Ree M,Kim K,Woo SH,Chang H(1997)Structure,chain orientation,and properties in thinfilms ofaromatic polyimides with various chain rigidities.J Appl Phys81:698–70821.Beck VA,Shaqfeh ESG(2007)Ergodicity-breaking and the unraveling dynamics of a polymer inlinear and nonlinear extensionalflows.J Rheol51:561–57422.Hlaing H,Lu XH,Hofmann T,Yager KG,Black CT,Ocko BM(2011)Nanoimprint-inducedmolecular orientation in semiconducting polymer nanostructures.ACS Nano5:7532–7538123 North China Institute of Aerospace Engineering (60.10.130.168) - 2014/1/6 Download。

matlab天线超表面代码以下是一个示例的MATLAB代码，用于模拟天线超表面的工作原理：matlab.% 定义超表面参数。

n = 10; % 超表面单元格的行数和列数。

a = 0.5; % 超表面单元格的边长。

lambda = 1; % 波长。

% 定义天线参数。

d = lambda/2; % 天线到超表面的距离。

% 定义场景参数。

L = 10lambda; % 场景尺寸。

% 创建场景。

x = linspace(-L/2, L/2, 100);y = linspace(-L/2, L/2, 100);[X, Y] = meshgrid(x, y);Z = zeros(size(X));% 计算超表面的相位调控。

k = 2pi/lambda; % 波数。

theta = atan2(Y, X); % 角度。

phase = exp(1ikdcos(theta)); % 相位调控。

% 计算场景中的电场分布。

E = phase.exp(1ikZ); % 电场分布。

% 绘制电场分布。

figure;imagesc(x, y, abs(E));axis square;colormap('hot');colorbar;title('电场分布');% 计算场景中的辐射模式。

P = abs(E).^2; % 辐射强度。

% 绘制辐射模式。

figure;polarplot(theta(:), P(:));title('辐射模式');这段代码演示了如何使用MATLAB模拟天线超表面的工作原理。

首先定义了超表面的参数，包括单元格的行数和列数、边长以及波长。

然后定义了天线的参数，包括到超表面的距离。

接下来创建了一个场景，包括场景的尺寸和坐标。

然后计算了超表面的相位调控，根据天线到超表面的距离和角度计算相位。

然后根据相位调控计算了场景中的电场分布，并绘制了电场分布图。

最后计算了场景中的辐射模式，并绘制了辐射模式图。