Particle Correlations at RHIC - Scrutiny of a Puzzle

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Mathematical model for heat transfer mechanism

Mathematical model for heat transfermechanism for particulate systemA.R.Khan *,A.ElkamelDepartment of Chemical Engineering,Faculty of Engineering and Petroleum,Kuwait University,P.O.Box5969,13060Safat,KuwaitAbstractVarious theoretical models for ﬂuidized bed to surface heat transfer have been considered to explain the mechanism of heat transport.The particulate ﬂuidized bed which is the common case for liquid–solid ﬂuidized bed is much simpler and homoge-neous and transport operation can be easily modeled.The heat transfer coeﬃcient in-creases to a maximum and then steadily decreases as the bed void fraction increases from that of a packed bed to unity.The void fraction e max at which the maximum value of heat transfer coeﬃcient occurs is a function of the solid–liquid system properties.An unsteady state thermal conduction model is suggested to describe the heat transfer process.The model consists of strings of the particles with entrained liquid moving parallel to surface,during the time interval heat conduction takes place.These strings are separated by liquid into which the principal mode of transfer is by convec-tion.The model shows a dependence of heat transfer coeﬃcient on void fraction and on physical properties,which is consistent with the results of experimental work.Ó2002Elsevier Science Inc.All rights reserved.1.IntroductionThe high heat transfer coeﬃcient from a hot surface to a ﬂuidized bed fa-cilitates the addition and removal of heat to and from a process eﬃciently.The role of various parameters and the mechanism of heat transfer in this ﬁeld have been the subject of extensive investigations during the last three decades.Many Applied Mathematics and Computation 129(2002)295–316*Corresponding author.Tel.:+965-481-7662;fax:+965-483-9498.E-mail address:rehman@.kw (A.R.Khan).0096-3003/02/$-see front matter Ó2002Elsevier Science Inc.All rights reserved.PII:S 0096-3003(01)00039-X296 A.R.Khan,A.Elkamel/put.129(2002)295–316A.R.Khan,A.Elkamel/put.129(2002)295–316297empirical correlations relating bed to surface heat transfer coeﬃcients for a range of operating variables have been proposed.They are of restrictive va-lidity because they cannot make adequate allowance for diﬀerent geometries of equipment used and varying degree of accuracy of the experimental techniques used.Furthermore,it is diﬃcult to extrapolate outside the experimental range of variables studied.Diﬀerent models have been proposed to explain the dif-ferent aspects of this complexproblem.There are particularly diverse concepts suggested by diﬀerent workers regarding the mechanism of heat transfer be-tween aﬂuidized bed and a heat transfer surface.In this paper an attempt has been made to explain the mechanism of heat transfer from bed to surface in liquidﬂuidized systems.The model results are compared with experimental data[1]and computed values of the existing models.2.Heat transfer models forﬂuidized bedBoth gas,and to a lesser extent liquidﬂuidized beds have been employed in chemical engineering practice particularly where the addition or removal of heat from the bed is required.In the case of gasﬂuidized beds the more important aspects have been collected and presented in detail by Zabrodsky [2].Most of the workers have examined a limited range of experimental variables and presented their results in the form of correlations;the power of any group in the correlations gave some indication of its importance within the experimental range investigated.It is clear that the scale of the equipment in which measurements were made has inﬂuenced the results.There is no suﬃciently general theory of heat transfer inﬂuidized beds,although several diﬀerent models have been proposed to explain various aspects of this problem.A brief description of some of the models is given in the following section.2.1.The limiting laminar layer modelLeva and Grummer[3]noted that the core of the bed was isothermal and oﬀered negligible thermal resistance while the main resistance limiting the rate of heat transfer between the bed and the heat source lay in aﬂuidﬁlm near the hot surface.They suggested that particles acted as turbulence promoters,which eroded theﬁlm reducing its resistive eﬀects.Levenspiel and Walton[4]derived an expression in terms of modiﬁed Nusselt and Reynolds numbers for the ef-fectiveﬂuidﬁlm thickness assuming that theﬁlm breaks whenever a particle touches the transfer surface.They have to modify the coeﬃcients in the model to account for their own experimental data.Wen and Leva[5]correlated the published heat transfer results on the basis of a scouring action model in which particle movement was assumed to be vertical and parallel to the heat source.Nu¼consC S qSd1:5Pﬃﬃﬃg pk g0:4GdPElgR"#0:36:ð1ÞIn this correlation theﬂuidization parameters are deﬁned as follows:1.E is theﬂuidization eﬃciencyðGÀG mfÞ=G mf.2.R is the expansion ration of the bed H=H mf.Richardson and Mitson[6]and Richardson and Smith[7]reported that for liquidﬂuidized beds the resistance to heat transport lay near the tube wall within the laminar sub-layer where the eﬀective thickness is reduced by the presence of particles for two reasons:1.The particles cause turbulence in theﬂuid thereby reducing the thickness ofthe laminar boundary layer.2.The particles themselves transport heat as a result of the radial componentof their rapid oscillating motion.Wasan and Ahluwalia[8]proposed that heat transfer through aﬂuidﬁlm was promoted byﬂuid eddies beyond theﬁlm boundary.They assumed that the solid particles were stationary and equally spaced and that heat was transferred through theﬁlm and then spread byﬂuid convection into the bulk of the bed.They compared the experimental results of various workers and found deviations of up toÆ44%.These models basically involve a steady-state concept of the heat transfer but Wasan and Ahluwala[8]included some dynamics transfer features for transfer through theﬂuid into the bed.298 A.R.Khan,A.Elkamel/put.129(2002)295–3162.2.Two resistanceﬁlm modelWasmund and Smith[9]suggested a modiﬁed laminar layer model,in which they considered particle convective transfer due to radial motion of particles into the laminar layer.This mechanism contributed50–60%of the total heat transferred and the remainder was fromﬂuid convective transfer.Tripathi et al.[10]used the series model proposed by Ranz[11]for eﬀective transport properties in packed beds.They compared results obtained by Wasmund and Smith[12]using radial velocities of the particles and observed deviation of Æ20%.Brea and Hamilton[13]and Patel and Simpson[14]used a two resis-tanceﬁlm model and emphasized that theﬂuid eddy convection is the main contributing factor to the heat transfer.Zahavi[15]measured the eﬀective diﬀusivity of theﬂuidized beds and also developed a semi-empirical correlation, which represented his results with a maximum deviation ofÆ34%.2.3.Unsteady state heat transferMickley and Trilling[16]suggested that the heat transfer process in a gas-ﬂuidized bed was of an unsteady state ter Mickley and Fairbanks[17] developed a model of heat transfer on the assumption that at any time there is unsteady state heat transfer within theﬂuidized bed close to heat source;this can be broken down into components due to solid/solid,solid/surface,gas/solid, and gas/surface transfer.Packets of loosely locked particles which are assumed to have uniform thermal properties constitute theﬂuidized bed.The mean heat transfer coeﬃcient can be calculated for packets of particles moving with a constant speed u passing rapidly along the length of the heat source,thenh¼2ﬃﬃﬃp pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃk0q0C0Pu0L r:Thermal conductivity k0,density q0and heat capacity C0P for packets can beestimated by use of the Schumann and Voss[18]correlation.The assumption that the thermal properties of the bed are uniform in the neighborhood of the heat source is unrealistic when the source and the bulk of the bed are at considerably diﬀerent temperatures.Mickley and Fairbanks[17]calculated the residence times of packets from resistanceﬂuctuations recorded for a thin electrically heated platinum strip.The frequency of packets was of the order of two per seconds and the residence time of0.4s.Henwood[19],Catipovic et al.[20],Suarez et al.[21]and George and Smalley[22]used a small heat transfer surface to measure the variations in local heat transfer coeﬃcient within and adjacent to a rising bubble.They concluded that heat transfer took place mainly through theﬂuid to the particle with the maximum rate in the vicinity of the contact part.A.R.Khan,A.Elkamel/put.129(2002)295–316299300 A.R.Khan,A.Elkamel/put.129(2002)295–3162.4.Simpliﬁed modelsThe heterogeneity ofﬂuidized beds is an important factor enhancing the value of the heat transfer coeﬃcient up to50–100folds for a gas and5–8folds for liquidﬂuidized bed.Botterill and Williams[23]have proposed a model for heat transfer in gas-ﬂuidized systems which are based upon the unsteady state conduction of heat to spherical particles adjacent to the transfer surface.The convective transfer through the gas is ignored because the eﬀective diﬀusivity of theﬂuidized bed is much higher than the eddy diﬀusivity of the gas.The particle and gas,whose temperatures were initially the same,approached the surface,the temperature of which remained approximately constant because of its high heat capacity.The Fourier equations for thermal conduction were solved byﬁnite diﬀerence technique.Apart from the axes of symmetry through the particles,there were three space limits to the problem.(A)Close to the heater it was assumed that there was a continuous thin layer ofpureﬂuid with which a particle was in contact.(B)The temperature of the other end the particle was set at the sink temper-ature,taken as the bulk temperature.(C)Transfer of heat between particles in a direction parallel to the surfacecould be neglected because the temperature diﬀerence between particles in adjacent position was very small.The experimental results for heat transfer coeﬃcients for the shortest resi-dence time of metallic particles were far less than the predicted values.This discrepancy was accounted for by assuming that a gasﬁlm of thickness equals to about10%of the particles around the heat source.Botterill[24]tried both models in which there was triangular and square packing on the particles and a ﬂuidﬁlm between the particles and the heat transfer surface.The thickness of theﬁlm was related to the resistance limiting the heat transfer.Diﬀerent workers made observations but no satisfactory conclusion was reached about the local variation of void fraction in the vicinity of the heat transfer surface.Davies[25]considered the unsteady state heat transfer by conduction be-tween the element at one temperature and spherical particles immersed in a liquid at another temperature.The particles were assumed to enter the thermal boundary layer with a mean radial velocity v r,approached the surface and left it again with the same velocity.The Fourier equations were solved by using the explicitﬁnite diﬀerence method and the same boundary condition as reported by Botterill and Williams[23].The very low values of particle convective heat transfer component predicted by the model indicated that only a very small proportion of the heat transferred from the heated surface to the bed was carried byﬂuidized particles.The experimental high value was attributable toA.R.Khan,A.Elkamel/put.129(2002)295–316301 the fact that the eﬀective thickness of the thermal boundary layer had been substantially reduced;this was mainly due to the following causes:1.The scouring action of particles.2.The high interstitial velocity of the liquid.2.5.Particle replacement modelGabor[26]has proposed that heat has been absorbed by the particulate bed based on string of spheres of inﬁnite length normal to heat transfer surface. Another simpliﬁed approach based on series of alternating gas and solid slabs also provided similar results as the spherical model.Gelperin and Einstein[27]have developed a more reﬁned model taking into account other details of the process involved.They considered that heat is transferred from the heat transfer surface by packets of solid particles by gas bubbles and by gas passing between the packet and the surface.The total heat transfer coeﬃcient is expressed ash¼h PðÀh convÞ1ðÀf0Þþh b f0þh r;ð2Þwhere h P;h conv;h b and h r are the heat transfer coeﬃcients corresponding to packet,convection,bubble and radiation,respectively,and f0is the fraction of time for which the surface is covered by bubbles.They solved the basic equations for their models of bed to surface heat transfer in terms of two heat resistances:R WS the resistance oﬀered by gas entrained by the particles close to the transfer surface and R a the resistance oﬀered by the gas–solid packets.They have tabulated theirﬁnal equations for instantaneous and mean heat transfer coeﬃcient for diﬀerent boundaries in their publication[27].For isothermal conditions of heat source,which have already been proposed a simpliﬁed solution can be used with little error.Martin[28]has presented a particle convective energy transfer model for wall to bed heat transfer from solid surface immersed in gas-ﬂuidized bed.In his model the following assumptions were applied.1.The contact time is regarded to be proportional to the time taken to coverthe path with the length of one particle diameter in freeﬂight.2.The wall to particle heat transfer coeﬃcient is calculated by integrating thelocal conduction heatﬂuxes across the gaseous gap between sphere and plain surface over the whole projected area of the sphere[29].3.The average kinetic energy for the random motion of particles comes from acorresponding potential energy.302 A.R.Khan,A.Elkamel/put.129(2002)295–3163.Development of mathematical modelFor the purpose of establishing a simpliﬁed model of a liquidﬂuidized bed, the system is assumed to consist of strings of particles with liquidﬁlling the intervening spaces.It is proposed that unsteady state thermal conduction takes place into both the liquid and the solid particles in the string as reported by Gabor[26].Liquid layers into which the principal mode of heat transfer is forced convection as shown in Fig.1to separate the strings.The overall heat transfer coeﬃcient for liquidﬂuidized bed from immersed surface constitutes solid conductive,liquid conductive and liquidconvective Array Fig.1.Mathematical representation ofﬂuidized particle entrained in liquid with initial and boundary conditions.components based on the void fraction determined by bed expansion charac-teristics and particle axial velocity V P;Axial.The convective component is cal-culated for liquid moving with interstitial velocity parallel to the heating surface.The conductive components for solid and liquid are evaluated based on contact time using unsteady state conduction equations for string of par-ticles with entrained liquid.3.1.Heat transfer across incompressible boundary layersThe simulated element is considered as aﬂat plate located on the axis of the column with the large faces parallel to liquidﬂow.The liquid with an average velocity u and uniform temperature T B passes over the hotﬂat plate at a constant temperature T E.At high Prandtl numbers the thermal boundary layer is always conﬁned entirely within the laminar sublayer.This limiting case of forced convection across a turbulent boundary layer can be solved analytically. Kestin and Persen[30]based their analysis on the laminar form of the energy equation and conﬁned their attention to the laminar sublayer only.The other assumption made is that the velocity varies linearly with distance perpendicular to theﬂat plate.The detailed solution of the energy equation for laminar form assuming linear velocity proﬁle within laminar sublayer is expressed asy s WðxÞlo To xÀy2d s Wd xo To y¼ao2To y2ð3aÞby substitutingg¼y3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðs W=lÞ3q9aR xx0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðs W=lÞpdx;Eq.(3a)can be transformed to ordinary diﬀerential equation.d2T d g2þgþ23d Td g¼0;ð3bÞwhereT¼T EÀTT EÀT B¼1;for x¼0and all values of y>0,T¼1,for y¼/and all values of x>0and T¼0,for y¼0and all values of x>0.The solution is given in the form of incomplete c-function asTðgÞ¼c1=3;g ðÞCð1=3Þ:The calculated values of the heat transfer coeﬃcient are presented together with experimental values[1]in Table2.The experimental data are obtainedA.R.Khan,A.Elkamel/put.129(2002)295–316303304 A.R.Khan,A.Elkamel/put.129(2002)295–316from immersed electrically wound heating surface in6-mm glass particleﬂui-dized bed dimethyl phthalate.The high experimental values are due mainly to the following causes:1.The plane element was not a trueﬂat plate.2.The edge eﬀects of the element caused a high value.3.The temperature of the element might not be uniform.In the case of aﬂuidized bed the presence of solid particles decreased the free area available forﬂow and caused an increase in the liquid velocity near the heat source.The contribution of heat transfer due to liquid alone was assessed on the basis of the interstitial velocity which was the factor determining the thickness of the thermal boundary layer.3.2.The contribution ofﬂuidized particles3.2.1.Particle velocities inﬂuidized bedsBy means of high speed photography several workers have measured the paths taken by a tracer particle in a transparentﬂuidized bed.Toomey and Johnstone[31]obtained particle velocities near the wall of dense phase gas-ﬂuidized bed.For the particular case of0.376-mm diameter glass spheresﬂu-idized in air,they reported particle velocities from60to600mm/s.Kondukov et al.[32]used radioactive tracer particles and radiation detectors to measure the particle trajectories in an airﬂuidized bed.Their results were similar to those of Toomey and Johnstone[31]and gave additional information on particle behavior in the interior of the bed.Handley[33]ﬂuidized1.1-and1.53-mm glass spheres with methyl benzoate in31-and76-mm diameter columns.He reported the radial and axial velocities of the particles for bed voidage ranging from0.67to0.905.He concluded that the velocity of the particles was completely random in a uniformlyﬂuidized bed.A more extensive study of particle velocities inﬂuidized bed was made by Carlos[34]and by Latif[35].Carlosﬂuidized9mm glass beads with dimethyl phthalate at30°C in a100mm diameter column.He reported the particle velocities(radial,angular,axial,horizontal and total)over the voidage range 0.53–0.7.A set of diﬀerential equations governing the mixing process was numerically solved by computer taking into account the eﬀect of radial and axial tif[35]extended the work of Carlos to6-mm glass spheres and developed a simple relationship between particle velocity and axial and radial positions.The constants of these equations are listed in Table1.These equations are used to evaluate particle velocities as a function of bed expansion characteristics.3.2.2.Residence time of particle in the vicinity of hot surfaceDavies [25]assumed that the high heat transfer coeﬃcient for the ﬂuidized bed was attributable to eﬀects arising from the radial velocity of the particles as reported by Figliola and Beasley [36].His model predicted very small values of heat transfer coeﬃcient because the thermal boundary layer thickness was small compared with the diameter of the particle and the residence time in the thin boundary layer was short.However from the average calculated values of radial and axial particle velocity components at the center of the ﬂuidized bed,it was obvious that the axial component of velocity was dominant.On this basis it appeared reason-able to assume,as a ﬁrst approximation,that particles and ﬂuid both at the bulk temperature of the bed approached the heat transfer surface at a velocity approximately equal to the average axial component of a particle velocity.The string of particles separated by intervening liquid thus traveled parallel to the hot surface and unsteady state heat conduction took place through the solid and liquid in parallel.The residence time for a particle could be given ast ¼L V P ;Axial:3.2.3.Unsteady state thermal conduction for liquid and solidIn the present model in which it is assumed that heat transfer is because of unsteady state thermal conduction in both liquid and solid,the following assumptions are made:1.The temperatures of the bulk of the ﬂuidized bed ðT B Þand the temperature of the element ðT E Þare uniform.2.A string of particles with entrained liquid at a uniform temperature equal to T B arrives quickly in an axial direction at the element.Table 1Coeﬃcients for calculation of axial velocity component [35]V P ;Axial ¼A Ár þB where r is normalized radial coordinate of particle A ¼a 0þa 1z þa 2z 2þa 3e a 4z2B ¼b 0þb 1z þb 2z 2þb 3e b 4z2e a 0a 1a 2a 3a 4b 0b 1b 2b 3b 40.55)0.767.25)8.74)87.4)20.20.42)3.08 3.8331.8)21.90.65)2.749.66)6.92)74.22)10.09 1.17)1.550.3846.5)17.030.75)1.42 4.78)2.84)128.2)15.110.96)3.17 1.9474.56)18.340.850.55)3.3 2.58)162.3)20.0)0.18 3.12)2.9286.7)22.70.95)4.27)17.9822.27)222.0)27.52.2212.04)15.09119.7)29.0A.R.Khan,A.Elkamel /put.129(2002)295–3163053.The particles and entrained liquid absorb heat by unsteady state conductionas they travel along the surface.Immediately the particles leave the vicinity of the element they exchange heat with the surrounding liquid.The model explains the way in which theﬂuidized particles contribute to-wards the transfer of heat between the hot surface and the bed.At any time the heat contents of both the liquid and the particles may be obtained from a knowledge of the temperature distribution within the particle and liquid.The temperature distribution in the solid and liquid in contact with the surface may be obtained by the heat conduction equation for both the liquid and solid over the residence time for which they are present at the hot surface.The heat conduction equation in spherical coordinates necessitates the use of three space dimensions;this may cause complications in solving the equation with its ap-propriate boundary conditions.The situation may be simpliﬁed by deﬁning the system in terms of Cartesian coordinates and assuming that the particles may be replaced by cubes,the length of the side of each of which is equal to the diameter of the particle.Each cube moves with one face in contact with the surface and liquid occupies the intervening spaces.Symmetry is assumed along the plane perpendicular to the surface.The length of the liquid slug between the particles and the thickness of the liquid layer separating the strings will be calculated as follows: In Fig.1a cube of dimensionðX SþX LÞis considered and the void fraction in the vicinity of the surface assumed to be the same as in the bulk where X S and X L are dimensions of particle and liquid slug,respectively.The volume ofthe particle is X3S and of the liquid slug X2SX L and void fraction is expressed ase¼ðX SþX LÞ3ÀX3S ðX SþX LÞ:On rearranginge¼1ÀX SX SþX L 3orX L¼X S11Àe1=3"À1#:The Fourier equations for unsteady state thermal conduction within the two homogeneous phases of the system areFor solid phaseo2T P o x2þo2T Po y2¼1a Po T Po t:ð4aÞ306 A.R.Khan,A.Elkamel/put.129(2002)295–316For liquid phaseo2T f o x2þo2T fo y2¼1a fo T fo t:ð4bÞThese equations are put in dimensionless form by deﬁningT P¼T EÀT PT EÀT B;T f¼T EÀT fT EÀT Band s¼tÁa P;s¼tÁa fo2T P o x2þo2T Po y2¼o T Po s;ð4cÞo2T f o x2þo2T fo y2¼o T fo s:ð4dÞInitial and boundary conditions,at t¼0,the particle and liquid slug both are divided to give a mesh nÂn and all points within the particle and liquid slug are at the bulk temperatureT P¼T f¼1;T P¼T f¼T B:ð5aÞThe temperature at the face of the liquid slug at x¼0and at all distances in the y-direction perpendicular to the surface is considered to be at the bulk temperature.The temperature of the similar face of the solid particle is taken as the computed values of the liquid temperature at the end of theﬁrst time step at x¼X Lat y¼0and x¼0;T f¼1;T f¼T B;T P¼T f and T P¼T fðAfter first time step at x¼X LÞ:ð5bÞAt t P0,the faces of both particle and liquid which are in contact with the surface are at all times at the surface temperature.At y¼0;T P¼T f¼0;T P¼T f¼T E:ð5cÞEqs.(4c)and(4d)are solved simultaneously usingﬁnite diﬀerence techniques for aﬁxed and for a variable boundary.For t>0theﬁxed boundary is ex-pressed by Eq.(5c)and the variable boundary is aty¼0at x¼0for liquidT f¼T P and T f¼T PðAfter previous time step at x¼X SÞand for particleA.R.Khan,A.Elkamel/put.129(2002)295–316307T P¼T f and T P¼T fðAfter current time step at x¼X LÞ:ð5dÞThe explicit method used by Botterill and Williams[23]and by Davies[25] restricts the time and space increments to ensure stability according to the equationD s61ðD XÞÀ2þðD YÞÀ2 h i:Their diﬀerence equation isTði;jÞkþ1ÀTði;jÞkD s ¼TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞD X2kþTði;jÀ1ÞÀ2Tði;jÞþTði;jþ1ÞD Y2k:ð6aÞThe above-mentioned method is very sensitive to the value of the operator k h which is given ask hX¼a hD tD X2and k hY¼a hD tD Y2:For the sake of simplicity equal increments are taken in the X-and Y-directions,i.e.D X¼D Y.An implicit method can make the equations inde-pendent of the operator value as well as of space and time increments.The diﬀerence equation is thenTði;jÞkþ1ÀTði;jÞkD s ¼TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞD X2þTði;jÀ1ÞÀ2Tði;jÞþTði;jþ1ÞD Y2kþ1:ð6bÞOn rearranging one gets a pentadiagonal matrixÀTðiÀ1;jÞÀTðiþ1;jÞþ1k hþ4Tði;jÞÀTði;jÀ1ÞÀTði;jþ1Þkþ1¼1k hTði;jÞk;ð6cÞwherek h¼a h D t D X2;which can be solved by either the Gaussian elimination method or the Gauss-Seidel iterative method to giveﬁve unknowns.The implicit alternating direction method discussed by Carnhan et al.[37], which avoids all the disadvantages discussed above,is thought to be most suited for this type of problem.The two diﬀerence equations are used in turn over successive time steps,each of duration D s=2.Theﬁrst Eq.(7a)is implicit 308 A.R.Khan,A.Elkamel/put.129(2002)295–316。

高能重离子碰撞

当两个高能核发生碰撞时，相互作用区域会发射许多粒子。

由于全同粒子的交换对称性，发射出的全同粒子具有玻色-爱因斯坦关联，又称HBT关联。

利用全同粒子携带的信息可以测量碰撞区域时空信息和相干性。

正反粒子的背对背关联(Back-to-Back Correlations)，简称BB 关联，与一对动量相反的粒子有关，它的出现是由于高密度发射源内的粒子质量位移。

本文用量子力学的波函数法推导了两粒子关联函数，并利用量子场论的知识研究了含质量位移效应的背对背关联和玻色-爱因斯坦关联函数。

由于实际的粒子发射源并非静态，而应该是随时间膨胀，考虑有限发射时间的影响，本文引入源的衰变随时间变化的分布，对含质量位移的HBT关联函数进行了修正。

这正是本文的创新点。

1. 高能重离子碰撞物理学1.1 高能重离子碰撞物理学简介在高能重离子碰撞以又称为高能核-核碰撞，通过高能重离子碰撞来产生极端高温度、高密度的核物质，研究产物的性质以寻找、探测可能存在的新物质相。

美国布鲁克海文实国家验室的相对论重离子对撞机RHIC和欧洲核子中心的大型强子对撞机LHC 都在做当前能量最高的相对论重离子碰撞实验。

1.2 相对论重离子碰撞的演化过程两核以较高能量碰撞时中心区域能量密度很高，靶核和入射核被高度激发后都会发生碎裂而产生了大量新粒子。

对高能核-核碰撞过程从时间上划分为四个阶段：初始阶段、压缩阶段、膨胀阶段、实验观察阶段。

1.3 夸克-胶子等离子体(QGP)自然界存在QGP的地方可能有两个，一是大爆炸后10μs左右的温度极高的初期宇宙；另一个则是重子数密度极高的中子星内部。

夸克被囚禁在强子内故不存在单个自由夸克。

QCD理论预测极高温度或极高密度下可能打破夸克禁闭形成“夸克—胶子等离子体”。

当前物理学存在两个谜题：夸克禁闭和破却的对称性，都有望在QGP 中得到解答。

1.4 强度干涉学强度干涉学最早是利用光子的强度干涉来测量星体的角径。

HBT关联与同时测量两个时空点上光子强度有关，关联程度依赖于发射源的角径。

PARTICLE ACCELERATOR

专利名称：PARTICLE ACCELERATOR发明人：LYONS, Bernard, John,SCHUETZ, Marlin, N.,VROOM, David, Archie申请号：EP93921695.0申请日：19930921公开号：EP0662238A1公开日：19950712专利内容由知识产权出版社提供摘要：A particle accelerator has a transmission window formed of a thin homogeneous foil having a predetermined thickness and having a predetermined length, and when laid fiat as a sheet having a transverse dimension. The window is formed to have a locus of a curve in cross section along the transverse dimension such that a radius of curvature of at least a portion of the curve in cross section is less than the length of the transverse dimension. Longitudinal channel and tubular shapes are preferred. Window cooling by gaseous and liquid fluid flows is also described. A transmission window assembly and a particle beam accelerator having an efficient rugged accelerator tube structure are also described. As one example, a liquid material processor and processing method employs either the curved window or a conventional window and advantageously directs liquid material onto the window to cool it, while a particle beam passing through the window enters the liquid material and changes it chemically in a predetermined manner. A mobile transporter enabling relocation of the liquid material processor between process sites is also described.申请人：RAYCHEM CORPORATION地址：300 Constitution Drive, Mail Stop 120/6600 Menlo Park, CA 94025-1164 US国籍：US代理机构：Jones, David Colin, et al 更多信息请下载全文后查看。

基于分子动力学研究刚性磨粒划擦铝基材料去除行为

第51卷第1期表面技术2022年1月 SURFACE TECHNOLOGY ·229·收稿日期：2020-11-29；修订日期：2021-05-03 Received ：2020-11-29；Revised ：2021-05-03基金项目：江苏省高等学校自然科学基金面上项目（19KJB430024）；江苏省工业软件工程技术研究开发中心开放基金重点项目（ZK190401）；南京工业职业技术大学国家自然科学基金培育项目（YK190109）Fund ：Supported by General Program of the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (19KJB430024), the Science Foundation of the Jiangsu Industrial Software Engineering Research Center (ZK190401), the Natural Science Foundation of Nanjing Vocational University of Industry Technology (YK190109) 作者简介：施渊吉（1989—），男，博士，副教授，主要研究方向为材料加工与表面技术。

Biography ：SHI Yuan-ji (1989—), Male, Doctor, Associate professor, Research focus: material processing engineering and surface treatment technology.引文格式：施渊吉, 程诚, 王捍天, 等. 基于分子动力学研究刚性磨粒划擦铝基材料去除行为[J]. 表面技术, 2022, 51(1): 229-239.SHI Yuan-ji, CHENG Cheng, WANG Han-tian, et al. Analysis of Material Remove Behavior Induced by Rigid Abrasive Particle for Aluminum 基于分子动力学研究刚性磨粒划擦铝基材料去除行为施渊吉1,2，程诚2，王捍天1，滕冰妍1，陈显冰1，何延辉1，张涛1，黎军顽3，郭训忠2（1.南京工业职业技术大学，南京 210046；2.南京航空航天大学材料科学与技术学院，南京 210016；3.上海大学材料科学与工程学院，上海 200072）摘要：目的实现材料高效去除，避免亚表层严重滑移，以及改善已划擦区表面形貌质量。

注射剂中的颗粒物英文788ParticulateMatter

<788> PARTICULATE MATTER IN INJECTIONSThis general chapter is harmonized with the corresponding texts of the European Pharmacopoeia and/or the Japanese Pharmacopoeia. These pharmacopeias have undertaken not to make any unilateral change to this harmonized chapter.Portions of the present general chapter text that are national USP text, and therefore notpart of the harmonized text, are marked with symbols () to specify this fact.Particulate matter in injections and parenteral infusions consists of mobile undissolved particles, other than gas bubbles, unintentionally present in the solutions.For the determination of particulate matter, two procedures, Method 1 (Light Obscuration Particle Count Test) and Method 2 (Microscopic Particle Count Test), are specified hereinafter. When examining injections and parenteral infusions for sub-visible particles Method 1 is preferably applied. However, it may be necessary to test some preparations by the light obscuration particle count test followed by the microscopic particle count test to reach a conclusion on conformance to the requirements.Not all parenteral preparations can be examined for sub-visible particles by one or both of these methods. When Method 1 is not applicable, e.g. in case of preparations having reduced clarity or increased viscosity, the test should be carried out according to Method 2. Emulsions, colloids, and liposomal preparations are examples. Similarly, products that produce air or gas bubbles when drawn into the sensor may also require microscopic particle count testing. If the viscosity of the preparation to be tested is sufficiently high so as to preclude its examination by either test method, a quantitative dilution with an appropriate diluent may be made to decrease viscosity, as necessary, to allow the analysis to be performed.The results obtained in examining a discrete unit or group of units for particulate matter cannot be extrapolated with certainty to other units that remain untested. Thus, statistically sound sampling plans must be developed if valid inferences are to be drawn from observed data to characterize the level of particulate matter in a large group of units.METHOD 1. LIGHT OBSCURATION PARTICLE COUNT TESTUse a suitable apparatus based on the principle of light blockage which allows an automatic determination of the size of particles and the number of particles according to size. The definition for particle-free water is provided in Reagent Specifications under Reagents, Indicators and Solution section.The apparatus is calibrated using dispersions of spherical particles of known sizes between 10 µm and 25 µm. These standard particles are dispersed in particle-free water.Care must be taken to avoid aggregation of particles during dispersion. System suitability can be verified by using the USP Particle Count RS (<11>).General precautionsThe test is carried out under conditions limiting particulate matter, preferably in a laminar-flow cabinet.Very carefully wash the glassware and filtration equipment used, except for the membrane filters, with a warm detergent solution and rinse with abundant amounts of water to remove all traces of detergent. Immediately before use, rinse the equipment from top to bottom, outside and then inside, with particle-free water.Take care not to introduce air bubbles into the preparation to be examined, especially when fractions of the preparation are being transferred to the container in which the determination is to be carried out.In order to check that the environment is suitable for the test, that the glassware is properly cleaned and that the water to be used is particle-free, the following test is carried out: determine the particulate matter in 5 samples of particle-free water, each of 5 ml, according to the method described below. If the number of particles of 10 µm or greater size exceeds 25 for the combined 25 ml, the precautions taken for the test are not sufficient. The preparatory steps must be repeated until the environment, glassware and water are suitable for the test.MethodMix the contents of the sample by slowly inverting the container 20 times successively. If necessary, cautiously remove the sealing closure. Clean the outer surfaces of the container opening using a jet of particle-free water and remove the closure, avoiding any contamination of the contents. Eliminate gas bubbles by appropriate measures such as allowing to stand for 2 min or sonicating.For large-volume parenterals, single units are tested. For small-volume parenterals less than 25 ml in volume, the contents of 10 or more units is combined in a cleaned container to obtain a volume of not less than 25 ml; the test solution may be prepared by mixing the contents of a suitable number of vials and diluting to 25 ml with particle-free water or with an appropriate particle-free solvent when particle-free water is not suitable. Small-volume parenterals having a volume of 25 ml or more may be tested individually. Powders for parenteral use are reconstituted with particle-free water or with an appropriate particle-free solvent when particle-free water is not suitable.The number of test specimens must be adequate to provide a statistically soundassessment. For large-volume parenterals or for small-volume parenterals having a volume of 25 ml or more, fewer than 10 units may be tested, based on an appropriate sampling plan.Remove four portions, each of not less than 5 ml, and count the number of particles equal to or greater than 10 µm and 25 µm. Disregard the result obtained for the first portion, and calculate the mean number of particles for the preparation to be examined.EvaluationFor preparations supplied in containers with a nominal volume of more than 100 ml, apply the criteria of test 1.A.For preparations supplied in containers with a nominal volume of less than 100 ml, apply the criteria of test 1.B.For preparations supplied in containers with a nominal volume of 100 ml, apply the criteria of test 1.B [Note: Test 1.A is used in the Japanese Pharmacopoeia]If the average number of particles exceeds the limits, test the preparation by the Microscopic Particle Count Test.Test 1.A — Solutions for parenteral infusion or solutions for injection supplied in containers with a nominal content of more than 100 mL.The preparation complies with the test if the average number of particles present in the units tested does not exceed 25 per mL equal to or greater than 10 µm and does not exceed 3 per mL equal to or greater than 25 µm.Test 1.B — Solutions for parenteral infusion or solutions for injection supplied in containers with a nominal content of less than 100 ml.The preparation complies with the test if the average number of particles present in the units tested does not exceed 6000 per container equal to or greater than 10 µm and does not exceed 600 per container equal to or greater than 25 µm.METHOD 2. MICROSCOPIC PARTICLE COUNT TESTUse a suitable binocular microscope, filter assembly for retaining particulate matter and membrane filter for examination.The microscope is equipped with an ocular micrometer calibrated with an objective micrometer, a mechanical stage capable of holding and traversing the entire filtration area of the membrane filter, two suitable illuminators to provide episcopic illumination in addition to oblique illumination, and is adjusted to 100 ± 10 magnifications.The ocular micrometer is a circular diameter graticule (see Figure 1) and consists of a large circle divided by crosshairs into quadrants, transparent and black reference circles 10 µm and 25 µm in diameter at 100 magnifications, and a linear scale graduated in 10 µm increments. It is calibrated using a stage micrometer that is certified by either a domestic or international standard institution. A relative error of the linear scale of the graticule within ± 2 per cent is acceptable. The large circle is designated the graticule field of view (GFOV).Two illuminators are required. One is an episcopic brightfield illuminator internal to the microscope, the other is an external, focusable auxiliary illuminator adjustable to give reflected oblique illumination at an angle of 10° to 20°.The filter assembly for retaining particulate matter consists of a filter holder made of glass or other suitable material, and is equipped with a vacuum source and a suitable membrane filter.The membrane filter is of suitable size, black or dark gray in color, non-gridded or gridded, and 1.0 µm or finer in nominal pore size.General precautionsThe test is carried out under conditions limiting particulate matter, preferably in a laminar-flow cabinet.Very carefully wash the glassware and filter assembly used, except for the membrane filter, with a warm detergent solution and rinse with abundant amounts of water to remove all traces of detergent. Immediately before use, rinse both sides of the membrane filter and the equipment from top to bottom, outside and then inside, with particle-free water.In order to check that the environment is suitable for the test, that the glassware and the membrane filter are properly cleaned and that the water to be used is particle-free, the following test is carried out: determine the particulate matter of a 50 ml volume of particle-free water according to the method described below. If more than 20 particles 10 µm or larger in size or if more than 5 particles 25 µm or larger in size are present within the filtration area, the precautions taken for the test are not sufficient. The preparatory steps must be repeated until the environment, glassware, membrane filter and water are suitable for the test.MethodMix the contents of the samples by slowly inverting the container 20 times successively. If necessary, cautiously remove the sealing closure. Clean the outer surfaces of the container opening using a jet of particle-free water and remove the closure, avoiding any contamination of the contents.For large-volume parenterals, single units are tested. For small-volume parenterals less than 25 ml in volume, the contents of 10 or more units is combined in a cleaned container; where justified and authorized, the test solution may be prepared by mixing the contents of a suitable number of vials and diluting to 25 ml with particle-free water or with an appropriate particle-free solvent when particle-free water is not suitable. Small-volume parenterals having a volume of 25 ml or more may be tested individually.Powders for parenteral use are constituted with particle-free water or with an appropriate particle-free solvent when particle-free water is not suitable.The number of test specimens must be adequate to provide a statistically sound assessment. For large-volume parenterals or for small-volume parenterals having a volume of 25 ml or more, fewer than 10 units may be tested, based on an appropriate sampling plan.Wet the inside of the filter holder fitted with the membrane filter with several milliliter of particle-free water. Transfer to the filtration funnel the total volume of a solution pool or of a single unit, and apply vacuum. If needed add stepwise a portion of the solution until the entire volume is filtered. After the last addition of solution, begin rinsing the inner walls of the filter holder by using a jet of particle-free water. Maintain the vacuum until the surface of the membrane filter is free from liquid. Place the membrane filter in a Petri dish and allow the membrane filter to air-dry with the cover slightly ajar. After the membrane filter has been dried, place the Petri dish on the stage of the microscope, scan the entire membrane filter under the reflected light from the illuminating device, and count the number of particles that are equal to or greater than 10 µm and the number of particles that are equal to or greater than 25 µm. Alternatively, partial membrane filter count and determination of the total membrane filter count by calculation is allowed. Calculate the mean number of particles for the preparation to be examined.The particle sizing process with the use of the circular diameter graticule is carried out by transforming mentally the image of each particle into a circle and then comparing it to the 10 µm and 25 µm graticule reference circles. Thereby the particles are not moved from their initial locations within the graticule field of view and are not superimposed on the reference circles for comparison. The inner diameter of the transparent graticule reference circles is used to size white and transparent particles, while dark particles are sized by using the outer diameter of the black opaque graticule reference circles.In performing the microscopic particle count test do not attempt to size or enumerate amorphous, semi-liquid, or otherwise morphologically indistinct materials that have the appearance of a stain or discoloration on the membrane filter. These materials show little or no surface relief and present a gelatinous or film-like appearance. In such cases the interpretation of enumeration may be aided by testing a sample of the solution by the light obscuration particle count test.EvaluationFor preparations supplied in containers with a nominal volume of more than 100 ml, apply the criteria of test 2.A.For preparations supplied in containers with a nominal volume of less than 100 ml, apply the criteria of test 2.B.For preparations supplied in containers with a nominal volume of 100 ml, apply the criteria of test 2.B. [Note: Test 2.A is used in the Japanese Pharmacopoeia]Test 2.A — Solutions for parenteral infusion or solutions for injection supplied in containers with a nominal content of more than 100 mL.The preparation complies with the test if the average number of particles present in the units tested does not exceed 12 per mL equal to or greater than 10 µm and does not exceed 2 per mL equal to or greater than 25 µm.Test 2.B — Solutions for parenteral infusion or solutions for injection supplied in containers with a nominal content of less than 100 mlThe preparation complies with the test if the average number of particles present in the units tested does not exceed 3000 per container equal to or greater than 10 µm and does not exceed 300 per container equal to or greater than 25 µm.。

Stochastic particle acceleration at shocks in the presence of braided magnetic fields

1. Introduction Diﬀusive particle acceleration at shock fronts has been advanced as an explanation for many astrophysical phenomena involving non thermal particles (for a review see Blandford & Eichler 1987). A key to the success of this theory is its simplicity: particles are assumed to diﬀuse in space upstream and downstream of a shock front, which is treated as a simple discontinuity in the velocity of the scattering centres. The well-known result is a power-law spectrum of accelerated particles f (p) ∝ p−sdiff with an index which is independent of the diﬀusion coeﬃcient, being determined solely by the compression ratio ρc of the shock front: sdiﬀ = 3ρc /(ρc − 1). In a recent Letter (Duﬀy et al. 1995) it was pointed out that the transport of particles is more complicated in a ‘braided’ magnetic ﬁeld, i.e., one which has a random component causing ﬁeld lines to wander around when projected onto the plane normal to their average direction. In particular, it was shown that there may exist a regime of sub-diﬀusive transport, in which the average square displacement of a particle from its origin grows with the square-root of elapsed time. This diﬀers sharply from the linear growth expected in the case of ordinary diﬀusion and reﬂects the restraining inﬂuence or trapping (in a stochastic sense) of charged particles by the magnetic ﬁeld. At shock fronts where particles must be transported across the direction of the mean magnetic ﬁeld in order to be accelerated, this leads to a modiﬁcation of the acceleration rate. In Sect. 2 we present a detailed discussion of the transport of particles in a braided ﬁeld, pointing out the diffusive and sub-diﬀusive regimes. There are two distinct physical eﬀects which come into play here: ‘microscopic’ diﬀusion i.e., that which would pertain in a uniform magnetic ﬁeld as a result of ﬂuctuations with a length scale approximately equal to the gyro radius of the particle concerned, and ‘macroscopic’ diﬀusion of the ﬁeld lines themselves, which results from ﬂuctuations of much longer length scale. The importance of braiding, or ‘ﬁeld line

大气颗粒物中硝酸盐和硫酸盐的来源及形成机制

当代化工研究Modern Chemical Research厶7 2020•23基础研究大气颗粒物中硝酸盐牙口硫酸盐的来源及形成机制*李婷何敏黄艺(成都理工大学地球科学学院四川610059)摘耍：随着经济的迅猛发展，城市大气环境质量越来越引起人们的关注，尤其是在大气中颗粒物的来源及生成途径等方面的研究.而硝酸盐和硫酸盐作为大气颗粒物中的重要组成部分，对大气环境、人体健康、大气能见度及气候变化等方面有着较大的影响.因此，查明大气颗粒物中硝酸盐和硫酸盐的形成途径对大气环境质量的管控具有重要的意义.关键词：硝酸盐；硫酸盐；来源；形成机制中阖分类号：0文献标识码：ASources and Formation Mechanism of Nitrate and Sulfate in Atmospheric Particulate MatterLi Ting,He Min,Huang Yi(School of Earth Sciences,Chengdu University of T echnology,Sichuan,610059) Abstracts With the rapid development of e conomy,more and more attention has been paid to the quality of u rban atmospheric environment. Especially the study on the source and generation of p articulate matter in the atmosphere.Nitrate and sulfate as important components of a tmospheric particles.It has a great impact on atmospheric environment,human health,atmospheric visibility and climate change.Therefore,it is of great significance to identify the f ormation pathways of n itrate and sulfate in atmospheric p articulates f or the control of a tmospheric environmental quality.Key words：nitrate；sulfate；source；formation pathways引言大气颗粒物成分复杂，对人体健康一叽大气能见度及气候变化间都有显著的影响，是雾霾的主要来源。

Quantum Transport Theory for Fusion Plasmas

Journal of Energy and Power Engineering 7 (2013) 1172-1177Quantum Transport Theory for Fusion PlasmasKeh-Ning Huang1, 2, Hsiao-Ling Sun2, Sheng-Fang Lin2 and Hao-Tse Shiao1, 21. Department of Physics, National Taiwan University, Taipei 10617, Taiwan, Republic of China2. Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan, Republic of ChinaReceived: June 13, 2012 / Accepted: November 19, 2012 / Published: June 30, 2013.Abstract: Fundamental quantum transport equation for impact-ionization processes in fusion plasmas is formulated in the actor-spectator description. The density-matrix formulism is adopted to treat both coherent and incoherent effects in a unified fashion. Quantum electrodynamic effects are also considered for high-temperature scenarios. Electron-impact ionization of uranium ion U91+ and proton-impact ionization of hydrogen are given as examples.Key words: Quantum collision, impact ionization, density matrix, QED (quantum electrodynamic) effects, fusion plasmas.1. IntroductionIn high-temperature fusion plasmas, accurate atomic and molecular collision data are needed for modelings. Besides fundamental interests, a viable quantum collision theory, which treats both electron-correlation and relativistic effects as well as recoils of colliding systems, is therefore necessary. The authors shall start from the relativistic equations of motion governing quantum collision processes in many-particle systems [1] and formulate the relativistic quantum transport theory in an ab initio manner. For collisions with heavy ions, QED (quantum electrodynamic) effects [2-4] may even double the collision cross section [5]. In the present formulation, the QED effects are however treated perturbatively as far as particle correlations are concerned.The conventional quantum collision theory for many-particle systems has kinematics and dynamics all entangled together. As an improvement from the outset, we shall separate kinematics and dynamics by a graphical method [4], which deals with angular-momentum couplings in an expedient andCorresponding author: Keh-Ning Huang, professor, research fields: transport phenomena, quantum collision, quantum electrodynamic theory and atomic structure. E-mail: ******************.tw.transparent manner. Because detailed polarization and angular correlations will also be treated in the present approach, we shall use the helicity-states [6] for both the incoming and outgoing channels. To include incoherent as well as coherent effects in a unified fashion, we shall adopt the density-matrix formulation [6].In cases of many-particle systems, particle correlations modify significantly the collision process. To demonstrate the importance of particle correlations in a many-particle system, let us imagine a person is such a system. The authors want to determine the properties of one particle in that person by the response of that particle to an external stimulus. Fig. 1 shows what would happen.A many-particle system is more like a breathing cell or a live person, in which all parts of the body are connected by nerve systems such that any outside stimulus acting on a small part of the body will cause a coordinated response of the entire person.As mentioned before, particle correlations are the most formidable effects to deal with when many-particle systems are involved. The authors however will look for a theory which just-describes the phenomenon being observed, not under-described as in the Newton’s equation of motion orover-described as in the Schrödinger equation for theAll Rights Reserved.Quantum Transport Theory for Fusion Plasmas1173Fig. 1 Actor-spectator description.entire many-particle system.So first let us see which particles of the system are observed and about what we will be making theoretical predictions, whereas the rest of the system will be considered as the heat bath or a heat reservoir with which the concerned particles contact. Those parts of the many-particle system which are observed or measured will be called the active particles while the rest called the spectator particles.Therefore, the authors shall employ the actor-spectator concept in the density-matrix formulation and always perform the ensemble average over spectator particles from the outset. Consequently, the resulting equations of motion after the ensemble average are at most for two active particles, because the present-day physics does not consider any operators beyond two-particle interactions. In cases where only one-particle excitations occur, we shall be dealing with equations of motion for one active particle only.Our equation of motion is in a sense the quantum-version of the Boltzmann transport equation. As a matter of fact, a variant of the present approach, called the MCRRPA (multi-configuration relativistic random-phase-approximation) has been applied to photoexcitation and photoionization processes with great success [7-14].In the present approach, all particle-correlations are treated in a truly ab initio manner including relativity, in which the coordinated response of the active particles is accounted for by performing the ensemble-average over spectator particles. That is, active particles are modulated by spectator particles such that the “bare” propagator for active particles becomes the “dressed” propagator. In other words, relativistic equations of motion are then derived for the “dressed” propagator. Bare and dressed propagators are presented symbolically in Fig. 2.For general applications, these equations may be derived in the second-quantization formalism in the quantum field theory; the authors shall present the derivation in a quantum-mechanical approach for intuitive understanding. The solution procedure of these equations is however quite involved, which includes resolving complicated angular-momentum couplings and irreducible spherical-tensor projections. Finally, coupled radial equations for active particles are obtained, and numerical procedures are in order. As demonstrations of the present approach, electron-impact ionization of uranium ion U91+ and proton-impact ionization of hydrogen will be presented.2. Collision EquationAny reaction, such as A + B + C + … E + F + G+ …, may be formulated by a collision equation as:(1)where denotes the final state of the products {E, F, G,…}, and denotes the initial state of the reactants {A, B, C,…}. Here represents the scattering-matrix of this reaction. In general, allkinematics and dynamics of the collision process are contained in the scattering matrix:(2) where represents generally a linear combination of one-and two-particle operators appropriate for the process.All Rights Reserved.Quantum Transport Theory for Fusion Plasmas1174Fig. 2 Bare and dressed propagators.3. Actor-Spectator Description ofMany-Particle SystemsIn the actor-spectator description, an N -particle system is formally devided into two parts: n active particles and N － nspectator particles. As mentioned earlier, the microscopic process for any physical phenomena may be summarized by an S-matrix,. In elaborate theories, both the initial-and final-states,and ,are described respectively as superpositions ofconfigurationsand constructedfrom orthonormal orbitals, while is a sum of one-particle operator and two-particle operator. Therefore, is in fact a summation over matrixelements of operators betweenconfigurationsand , denoted symbolically as:(3)Particles connected through the operatorin thefinal- and initial-configurations and aredefined as active particles, i.e., actors, and the rest ofthe many particles are called spectator particles, i.e.,the spectators. For each matrix element, it is obvious that spectator particles must stay in the same orbitals in the final- and initial-configurations otherwise the matrix element willvanish. Consequently, in the evaluation of, the integrations over N － n spectatorparticles can always be performed formally withoutknowing the explicit forms of orbitals occupied by spectator particles. For symmetric cases, operatorsandhave the general forms:(4)(5)4. Transition MatricesIn terms of N -particle wave functions, the authorshave the full transition matrix for the N -particle systemdefined as:(6)where . The reduced transition matrix, or called specifically the n th-order transition matrix, is in turn defined as:(7)wherewith intrinsic variables, such as spin, included implicitly. Here denotes taking the trace of the operator in any specific representation over k particle variables. The n th-order transition matrixis the “dressed” propagator forn active particles, and the “dressing” of the propagator for n active particles is carried out formally by taking ensemble average over N － n spectator particles.5. Evaluation of the S -MatrixFor single configurations with one active particle, the S -matrix is given as:(8) where for ease of notations, we adopt the convention that the operator operates only on variables with particle label 1 and not on those with particle label ,All Rights Reserved.Quantum Transport Theory for Fusion Plasmas1175but the integration overtreats both labels 1 andas if they were the same. It is clear that to evaluate the S -matrix with one active particle, it suffices to know the 1st-order transition matrix . Similarly, the S-matrix with two active particlesmay be evaluated as:(9)In summary, knowingis all that is needed to evaluate the S-matrix forany collision process.6. Relativistic Equations of Motion forTransition MatricesChoosingandto be energy eigenstates ofan N -particle system, we get:(10) (11)Consequently, the authors may easily derive(12)with. Here for ease of notations, theauthors suppress the superscript in, i .e .,. By taking ensemble average over spectatorparticles, the authors obtain in general:(13)This is our relativistic equation of motion for the n th-order transition matrix [1].The relativistic equation of motion for the first-order transition matrix is:(14)which may be given in time-independent form as:(15)Here and denote one-particle and two-particle operators in the total Hamiltonian of the N -particle system:(16)The relativistic equation of motion for the second-order transition matrix is:(17)which has the time-independent form:(18)7. ApplicationsThe election-impact ionization of the hydrogen-like U 91+ is described by the reaction:(19) The electromagnetic interaction between char-ged particles arising from the exchange of one photon may be summarized in the QED theory as the Coulomb interaction plus transverse-photon interaction [3]:(20)The QED cross sections of electron-impact ionization for hydrogen-like U 91+ have been calculated and are shown in Fig. 3. The results agree quite well with experiment [15, 16] and with available theoretical data [17, 18].The proton-impact ionization of hydrogen has been calculated in the present formulation, and the results are presented in Fig. 4. There is an excellent agreement with experiment [19-21].8. ConclusionA relativistic quantum transport theory for many-particle systems has been proposed to treat particle-correlation and relativistic effects in an ab initioAll Rights Reserved.Quantum Transport Theory for Fusion Plasmas1176Fig. 3 Electron-impact ionization of U91+.Fig. 4 Proton-impact ionization of hydrogen.manner. The ensemble average over spectator particles of the many-particle system is formally carried outfrom the outset to reduce the problem only to that ofactive particles. Important characteristics of this approach may be summarized as follows :(1) Applicable to both closed-shell and open-shellmany-particle systems; (2) Gauge independent for radiative transitions; (3) Fine structures and spin polarizations are built in;(4)Both discrete-state and continuum-statecorrelations are dealt with; (5) Multi-electron excitations are incorporated; (6) Initial- and final-state correlations are treated onan equal footing;(7) Core polarization and shielding effects are included self-consistently. An approximation to the present approach, called theMCRRPA (multiconfiguration relativistic random-phase approximation), has been applied to photoexcitation and photoionization with great success. We have further incorporated quantum electrodynamic effects perturbatively in the formulation. Our results for the electron-impact ionization of U 91+ agree well with existing experimental and theoretical data. The proton-impact-ionization cross sections of hydrogen are calculated including recoil effects and are in excellent agreement with available data.References[1] K.N. Huang, Relativistic many-body theory of atomictransitions. The relativistic equation-of-motion approach, Phys. Rev. A 26 (2) (1982) 734-739.[2] K.N. Huang, Calculation of the vacuum-polarizationpotential, Phys. Rev. A 14 (1976) 1311-1318.[3] K.N. Huang, Covariant photon interaction in atomicphysics, Phys. Rev. A 18 (1978) 1119-1123.[4] K.N. Huang, Graphical evaluation of relativistic matrixelements, Rev. of Mod. Phys. 51 (1979) 215-236.[5] H.L. Sun, J.C. Chang, J.T. Hsiao, S.F. Lin, K.N. Huang,Electron-impact ionization of hydrogenlike ions in QED theory, Phys. Rev. A 81 (2010) 042711.[6] K.N. Huang, Theory of angular distribution and spin polarization of photoelectrons, Phys. Rev. A 22 (1980) 223-239. [7] K.N. Huang, W.R. Johnson, Multiconfigurationrelativistic random-phase-approximation. Theory, Phys. Rev. A 25 (1982) 634-649.[8] W.R. Johnson, K.N. Huang, Resonance transitions ofbe-like ions from multiconfiguration relativistic random-phase approximation, Phys. Rev. Lett. 48 (1982)315-318. [9] H.C. Chi, K.N. Huang, K.T. Cheng, Autoionizing levels ofberyllium from the multiconfiguration relativisticrandom-phase approximation, Phys. Rev. 43 (1991) 2542-2545.[10] H.C. Chi, K.N. Huang, Photoionization of beryllium in the multiconfiguration relativistic random-phaseapproximation, Phys. Rev. A 43 (1991) 4742-4745.[11] K.N. Huang, H.C. Chi, H.S. Chou, The MCRRPA theory and its applications to photoexcitation and photoionization,Chin. J. Phys. 33 (1995) 565-644. [12] L.R. Wang, H.C. Chi, K.N. Huang, Spin-orbit andcore-shielding effects on double-excitation resonances ofZn, Phys. Rev. Lett. 83 (1999) 702-705. [13] L.R. Wang, J.T. Hsiao, K.N. Huang, Doubly excitedAll Rights Reserved.Quantum Transport Theory for Fusion Plasmas 1177states of the Be atom in the MCRRPA, J. Phys. B 39 (2006) ) L217-L224.[14]J.T. Hsiao, L.R. Wang, H.L. Sun, S.F. Lin, C.L. Lu, K.N.Huang, Photoionization processes of the beryllium atom,Phys. Rev. A 78 (2008) 013411-013419.[15]N. Claytor, B. Feinberg, H. Gould, C.E. Bemis, C.A.Ludemann, C.R. Vane, et al., Electron impact ionization of U88+-U91+, Phys. Rev. Lett. 61 (1988) 2081-2084. [16]R.E. Marrs, S.R. Elliott, D.A. Knapp, Production andtrapping of hydrogenlike and bare uranium ions in an electron beam ion trap, Phys. Rev. Lett. 72 (1994) 4082-4085.[17] D.L. Moores, K.J. Reed, Effect of the moller interactionon electron-impact ionization of high-z hydrogenlike ions,Phys. Rev. A 51 (1995) R9-R11.[18] C.J. Fontes, D.H. Sampson, H.L. Zhang, Fully relativisticcalculations of and fits to 1s ionization cross sections,Phys. Rev. A 59 (1999) 1329-1335.[19]M.B. Shah, H.B. Gilbody, Experimental study of theionisation of atomic hydrogen by fast H+ and He2+ ions, J.Phys. B 14 (1981) 2361-2377.[20]M.B. Shah, D.S. Elliott, H.B. Gilbody, Ionisation ofatomic hydrogen by 9-75 keV proton, J. Phys. B 20 (1987) 2481-2485.[21]M.B. Shah, J. Geddes, B.M. Mclaughlin, H.B. Gilbody,New low-energy measurements and calculation of ionization in H+-H collision, J. Phys. B 31 (1998) L757-L762.All Rights Reserved.。

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a r X i v :n u c l -t h /0209055v 1 19 S e p 20021Particle Correlations at RHIC –Scrutiny of a Puzzle Sven Soﬀa ,Steﬀen A.Bass b ,David H.Hardtke a and Sergey Y.Panitkin c a Lawrence Berkeley National Laboratory,Nuclear Science Division MS70-319,1Cyclotron Road,Berkeley,CA94720,USA b Department of Physics,Duke University,Durham,NC27708,USA,and RIKEN BNL Research Center,Brookhaven National Laboratory,Upton,NY11973,USA c Physics Department,Brookhaven National Laboratory,PO Box 5000,Upton,NY11973,USA We present calculations of two-pion and two-kaon correlation functions in relativistic heavy ion collisions from a relativistic transport model that includes explicitly a ﬁrst-order phase transition from a thermalized quark-gluon plasma to a hadron gas.We compare the obtained correlation radii with recent data from RHIC.The predicted R side radii agree with data while the R out and R long radii are overestimated.We also address the impact of in-medium modiﬁcations,for example,a broadening of the ρ-meson,on the correlation radii.In particular,the longitudinal correlation radius R long is reduced,improving the comparison to data.Two-particle correlations at small relative momenta have been predicted to be partic-ularly sensitive to a phase transition from quark-gluon matter to hadronic matter [1].For a ﬁrst-order phase transition,larger hadronization times were expected to lead to considerably enhanced correlation radii,characterizing the space-time extension of the particle-emitting source,compared to,for example,a purely hadronic scenario.The radii should also depend on the critical temperature T c ,the latent heat,the initial speciﬁc entropy density or the initial thermalization time of the quark-gluon phase.Here,we discuss relativistic transport calculations at RHIC energies that describe the initial dense stage by hydrodynamics [2]and the later more dilute stages by microscopic transport [3]of the particles.The two models are matched at the hadronization hyper-surface [4].In the hadronic phase the particles are allowed to rescatter and to excite resonances based on cross sections as measured in vacuum.For the initial dense (hydro-dynamical)phase a bag model equation of state exhibiting a ﬁrst-order phase transition is employed.Hence,a phase transition in local equilibrium that proceeds through the formation of a mixed phase,is considered.The details of this relativistic hybrid transport model can be found elsewhere [4].We ﬁrst brieﬂy summarize the main conclusions obtained in previous work [5,6,7].Then,we show in detail the results for pions when calculating explicitly the correlation functions from the source function of the transport model.These results are subsequently compared to calculations that take in-medium modiﬁcations into account and to experimental data2√for central Au+Au collisions at3K T (GeV/c).R O ,R S ,R L (f m ),*10K T (GeV/c)Figure 3.Pion correlation parameters R o ,R s ,R l ,and λas obtained from the ‘QGP+hadronic rescattering’transport model calculations +correlation after burner (T)compared to experimental data from PHENIX and STAR.For theory (T),the eﬀects of a ﬁnite momentum resolution (f.m.r.)and a broadened in-medium ρspectral function (ρi .m .)are calculated.Note the strong reduction of R l when an in-medium ρis considered.The pion correlation parameters R o ,R s ,R l and λare obtained from the explicit cal-culation of the correlation functions [11]in the respective transverse momentum bins and subsequent ﬁtting of these correlation functions to a Gaussian form of the correlatorC 2=1+λexp(−R 2o q 2o −R 2s q 2s −R 2l q 2l ).The results of ﬁtting the 3-dimensional pion cor-relation functions (as obtained from the transport calculations (RHIC initial conditions,T c ≈160MeV)+correlation after burner (by Pratt)[11])are shown,together with the experimental STAR and PHENIX data [8,9],in Figure 3.Even for a ﬁrst-order phase transition scenario,the calculated interferometry radii (T)are not unusually large.Fi-nite momentum resolution (f.m.r.)reduces the true correlation radii and the λintercept parameter,in particular,at higher K T [6].This is corrected for in the experimental anal-ysis.While the R side radii show agreement the R out radii are too large compared to the data.However,these pion radii are considerably smaller than the corresponding radii obtained from the coordinate-space points and using expressions for the Gaussian radius parameters based on a saddle-point integration over the source function [5,6,7,12].Only the calculations as shown in Figure 3which are obtained from the complete calculation and the performed ﬁts should be compared to data.The R out /R side ratio is also larger than unity for the ﬁtted values and conﬁrms the so-called HBT-puzzle ,i.e.,the RHIC data from STAR and PHENIX [8,9]indicate a decreasing ratio with K T (even below 1)4and the calculations show only ratios larger1.We remark that at the CERN-SPS(see Refs.in[5])the R out/R side ratio seems to increase with K T,being larger than1,im-plying a real qualitative change of the reaction dynamics from SPS to RHIC energies.A discussion of theoretical assumptions and uncertainties,experimental corrections,and possible solutions of this problem(including strongly opaque sources or scenarios with strong supercooling leading to spinodal instabilities)is provided in[5].In-medium modiﬁcations may alter the correlation radii due to their impact on the mean free path via modiﬁed cross sections atﬁnite temparatures or densities(T f.o.∼100−120MeV).Increasing the width of theρfrom the vacuum value ofΓρ=150MeVtoΓi.m.ρ=350MeV enhances the opacity of the system(moreππ↔ρprocesses),with aparticular reduction of the radius R long,improving the comparison to data(Fig.3).We have demonstrated that the kaon interferometry measurements,in particular at high K T,will provide an exellent probe of the space-time dynamics(close to the phase boundary).In addition they represent a severe test of the pion correlations and may help to better understand the HBT-puzzle[13],i.e.,the diﬀerence in the K T dependence of the R out/R side ratio between the model predictions and the experimental RHIC data.The diﬀerences are due to the R out radii(which are larger in the model calculations)while the R side radii seem to be described reasonably.An in-medium broadening of theρ-meson reduces in particular R long,improving the comparison to data.Acknowledgments:SS is supported by the Alexander von Humboldt Foundation through a Feodor Lynen Fellowship and DOE Grant No.DE-AC03-76SF00098. 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