Bose-condensation through resonance decay

a r X i v :n u c l -t h /9308015v 1 19 A u g 1993Bose-condensation through resonance decay

U.Ornik 1?,M.Pl¨u mer 2?and D.Strottman 3

?

1Theory Group,Gesellschaft f¨u r Schwerionenforschung (GSI),Darmstadt,F.R.Germany 2Physics Department,University of Marburg,FRG 3Theoretical Division,Los Alamos National Laboratory,Los Alamos,USA 87545February 9,2008Abstract We show that a system described by an equation of state which contains a high number of degrees of freedom (resonances)can create a considerable amount of super-?uid (condensed)pions through the decay of short-lived resonances,if baryon number and entropy are large and the dense matter decouples from chemical equilibrium ear-lier than from thermal equilibrium.The system cools down faster in the presence of a condensate,an e?ect that may partially compensate the enhancement of the lifetime expected in the case of quark-gluon-plasma formation.

M@VAX.HRZ.UNI-MARBURG.DE

?E.Mail:DDS@https://www.wendangku.net/doc/0117069567.html,

1

The investigation of hot hadronic matter through heavy ion collisions has shown that an understanding of medium e?ects and?nal state interactions is of great importance for the interpretation of the particle spectra in order to distinguish new physical phenomena like the formation of quark-gluon plasma(QGP)from background phenomena caused by “conventional”physical e?ects.We refer here,e.g.,to the strangeness enhancement,the J/Ψsuppression and the soft-pion puzzle,which can in principle be explained either by the formation of a QGP or from purely hadronic origin(for a recent review,see ref.[1]).

An interesting explanation for the soft pion puzzle was proposed in[2].The authors argued that the baryonic and mesonic resonances may decouple from chemical equilibrium but still remain in thermal equilibrium.Since the resonance production rates fall drastically after their decoupling from chemical equilibrium,the remaining short-lived resonances would then rapidly decay into pions.As a consequence,the pions may acquire a non-zero chemical potentialμπwhich leads to a softer p⊥-spectrum.

A di?erent possible explanation for the absence of local chemical equilibrium of pions in the?nal(decoupling)stage,which should not be confused with the scenario discussed in ref.[2],was proposed in refs.[3,4]where it is assumed that the pions are initially created out of local chemical(and even thermal!)equilibrium and it turns out that the system never reaches complete local chemical equilibrium[3,4].

In the present letter,we investigate the scenario of ref.[2],where it is assumed that the dense matter reaches local thermodynamic equilibrium in an early stage of the expansion. The authors of[2]considered only the case that the chemical potential stays below the pion mass.Moreover,they did not attempt to determine the value of the pion chemical potential in terms of the temperature and baryon number density at the point of decou-pling from local chemical equilibrium.Below,we address the problem if and under what conditions the decay of short-lived resonances can lead to the formation of a pionic Bose condensate,and what happens to the condensate in the subsequent expansion until the system decouples from local thermal equilibrium and the?nal state particles are emitted. To do so,we explicitly consider the transition from a resonance gas to a gas of stable and long-lived particles for a hadronic equation of state which contains the known resonances up to masses of2GeV,imposing constraints from the conservation of energy-momentum,

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baryon number and strangeness.In particular,we shall show that for a system rich in entropy and baryon numberμπmay reach the pion mass and,consequently,the pions may form a Bose condensate in the central region in relativistic heavy ion collisions.

For a hydrodynamically expanding system,the formation of such a condensate implies the presence of a super?uid component.The idea that the hadronic matter might be super?uid was put forward a long time ago in a somewhat di?erent context[5].This could explain in a natural way the presence of a coherent component in multiparticle production.The implications for pion interferometry measurements will be discussed in a separate paper[6].We note that although the creation of a remarkable amount of condensed pions is interesting by itself,these e?ects could also be used to answer the question of whether pions and resonances are in chemical equilibrium before they decouple from thermal equilibrium.Because the?nal stage in the hadronic phase can have a signi?cant in?uence on the particle spectrum,the question of whether the freeze-out occurs in chemical and thermal equilibrium,as assumed in[7,8]or only in thermal equilibrium[2]is of great importance.

In order to investigate the conditions necessary to form a condensate,let us?rst consider a system in local thermodynamic(i.e.,thermal and chemical)equilibrium.The medium is then completely described in terms of its equation of state(EOS)which can be expressed in the form

εfl=f(T fl,μk fl)(1)

whereεfl is the energy-density of the?uid,T fl the temperature,and the quantitiesμk fl are the chemical potentials related to conserved charges Q k(such as baryon number B and strangeness S).The index fl refers to quantities which describe the?uid.

During a heavy ion collision we expect under certain conditions the formation of a system of hot hadronic matter or a QGP described by eq.(1)which is in chemical and thermal equilibrium.The system then undergoes a phase of hydrodynamic expansion until it has become so dilute that the interactions are no longer strong enough to maintain local equi-

3

librium.This?nal stage,when the collective behaviour terminates and particles decouple from the dense matter,is referred to as freeze-out.

The scenario of ref.[2]di?ers from most of the other hydrodynamic descriptions of heavy ion collisions in so far as in[2]the dense matter is assumed to go out of chemical equilibrium but remain in thermal equilibrium for some time.Or,to put it di?erently,there will be a“chemical freeze-out”independent of and prior to the“thermal freeze-out”.As was mentioned above,the transition out of chemical equilibrium leads to the decay of short-lived resonances and the appearance of a non-zero chemical potential of the pions.

Below,we shall assume that the chemical freeze-out,just like the thermal freeze-out,occurs “instantaneously”(i.e.,on a three-dimensional hypersurface characterized by a condition such as T(x,t)=const.).This seems a reasonable assumption since the relevant time scales are the decay times of the short-lived resonances which are on the order of～1 fm/c.To illustrate the rapidness of such a transition,we consider the simpli?ed case of a system of pions and rho-mesons in local thermal equilibrium that undergoes a longitudinal scaling[9]expansion.In addition to the decayρ→ππwe also take into account the inverse processππ→ρ+X.The equations which describe the time evolution of the system are

dε

τ(2)

dnπ

τ+2

nρ

dτ=?

nρ

τρ

+ σv n2π(4)

whereε=ερ+επand P=Pρ+Pπare the energy density and pressure,n i(i=ρ,π) are the number densities andτ≡

√

transition is even faster.The?gure also shows that the decay of rho-mesons alone is not su?cient to drive the pions into the condensate(for that,one needs to take into account the decay contributions of all the other short-lived resonances as well).

We now return to the description of the expanding hot and dense matter at the point of chemical freeze-out.It will be assumed that the?uid freezes out into“stable”π’s,K’s, nucleons,Λ’s and the long living resonancesω’s andη’s.The term“stable”here means that the decay of the particle or resonance occurs only after the complete(thermal and chemical)freeze-out of the system.In particular,the contribution ofω’s andη’s to the pion chemical potential is zero.Nevertheless,their later decay leads to a non-thermal component of the pion spectrum which is determined by the decay kinematics[7].

The system of chemically frozen out particles can be described by a chemical decoupling temperature T ch.f.,chemical potentials for baryons and strange particles,μB andμS, and a pion chemical potential,μπ,which describes the overpopulation of pions due to the decoupling of resonances.The chemical potentials are determined by the requirement that the energy densityεfl,the baryonic density b fl and the strangeness density s fl=0of the ?uid are equal to those of the system after chemical freeze-out.We obtain the following system of equations:

εfl=εtherm

π(μπ,T ch.f.)+mπn con

π

Θ(mπ?μπ)

+

i=K,N,Λ,ω,η,...εtherm

i

(μS,μB,T ch.f.)(5)

b fl= k B k n therm k(μS,μB,μπ,T ch.f.)(6)

s fl= k S k n therm k(μS,μB,μπ,T ch.f.)=0(7) where

εtherm i (μS,μB,μπ,T)=

g i

exp E i??μi

2π2

p2dp1

T ±1

(9)

5

are the thermal parts of the energy density and the number density of particle species i, and where we have used the notation

?μi= k Q k iμk(10) for the chemical potentials.Note that on the r.h.s.of eq.(10)the charges Q k include the pion number Nπwhich is a conserved quantity after chemical freeze-out.

In eq.(5)we have introduced a term n con

π

that describes the condensed component of pions which will appear if the chemical equilibrium for pions reaches the pion mass.In this case an overpopulation of pion states occurs and the remaining pions will be forced into the ground state,thereby forming a Bose condensate.The?uid then consists of a thermal component and a super?uid component.The latter moves with the rapidity of the?uid, whereas the thermal pions have on top of the?uid rapidity a thermal distribution.The resultant single inclusive distribution of pions emitted from a hydrodynamically expanding source

1

dydp⊥=

gπ

exp p iμuμ??μπ

resonances and a treatment of compression e?ects,thus retaining the essential features of nuclear matter near the ground state[11].In addition to the work presented in[11]we take into account the e?ects of?nite baryonic and mesonic masses,use Fermi statistics for the baryons and include mesons up to masses of2GeV.

Fig.2shows the thermal and the condensate component of pions at chemical freeze-out, and the chemical potentialsμπ,μb andμs,as functions of the temperature T ch.f.that characterizes the decoupling from chemical equilibrium,for three di?erent values of the baryon number density b.It can be seen how the baryonic and strange chemical potentials,μb andμs,decrease with increasing temperature T ch.f..Clearly,the strange chemical potential induced by the?nite baryon number never execeeds150MeV.As the amount of heavier resonances increases with temperature,the pion chemical potentialμπgrows with T ch.f..It becomes equal to the pion mass in the temperature region170-200MeV;the value of T ch.f.whereμπ=mπdepends on the baryonic density.This behaviour is also re?ected in the dependence of the thermal and condensate densities of pions on the temperature and baryon number density at chemical freeze-out.It can be seen that at su?ciently high baryonic densities and temperatures the condensate contribution to the chemically decoupled pions becomes signi?cantly large.It is noteworthy that the condensation e?ect is enhanced if the density of the strange particles has not yet reached its equilibrium value. In this case the remaining energy is distributed among the non strange resonances,and this leads to an enhanced production of particles in the condensate.

Finally,we need to discuss the question whether or not the condensate survives the time period between the chemical and the thermal freeze-out.This is of particular importance if the dense matter decouples from chemical equilibrium at temperatures T ch.f.～170?200 MeV which are considerably higher than those usually associated with the thermal freeze-out,T th.f.～mπ.After decoupling from chemical equilibrium,the system of stable and long-lived particles continues to expand until it has cooled down to the temperature T th.f..

For a one-dimensional scaling expansion,the evolution of the system is determined by the equations

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s τ

=const .(12)b τ

=const .(13)s e τ

=const .(14)n πτ

=const .(15)n ατ=const .(16)

which describe the conservation of entropy S e ,baryon number B ,strangeness S and pion number N π(s e ,b ,s and n πbeing the corresponding densities,respectively).The index αlabels those of the stable particle species (K,N,Λ,ω,η,...)which also have decoupled from chemical equilibrium.For the gas of stable particles and long-lived resonances,the thermodynamic quantities are given by the expressions on the r.h.s.of eqs.(5)–(10),with s e =

i (εi +P i ??μi n i )/T .Energy density and number density of the pions consist

of a thermal and a condensate component,επ=εtherm π+m πn con πand n π=n therm π+n con π,respectively.Note that the condensate does not contribute to the entropy.Consequently,the cooling of the system –i.e.,the function T (τ)–does not depend on the super?uid component.Fig.3shows the cooling curves T (τ)of a pion gas,for di?erent values of the pion chemical potential.For comparison we have also included the results for an ideal gas of massless pions.It can be seen that the cooling rate increases with increasing pion chemical potential and becomes maximal in the presence of a condensate.This implies that Bose condensates may reduce the lifetime of the ?reball.

In particular,we are interested in the time dependence of the fraction of pions in the

condensate,f con ≡n con π/(n therm π+n con π

).Eqs.(12,15)imply that for the fraction of thermal pions,f therm =1?f con ,

f th.f.therm =f ch.f.therm n therm π/s e ) th.f.

the pion gas,it turns out that in the temperature range between200MeV and150MeV, f therm varies by less than10%.

To summarize,we have shown that the excited hadronic matter created in ultrarelativistic heavy ion collisions could form a pion condensate if(i)the system is rich in baryon number and entropy density,(ii)the hadrons decouple from local chemical equilibrium earlier than from thermal equilibrium,as suggested in[2],and(iii)the loss of chemical equilibrium occurs at temperatures of～180MeV or higher.The e?ect depends on the chemical decoupling temperature,on the baryonic density and,rather sensitively,on the resonance contribution to the EOS.1The e?ect is increased if the chemical equilibrium for strange particles is not complete,i.e.if the amount of strangeness which is initially zero and starts to increase during the collision and expansion process has not yet reached its equilibrium value.

It is not the purpose of the present study to prove that such a decoupling process occurs. To do so would require a detailed knowledge of the density and temperature dependence of the inelastic and elastic cross sections of hadrons in the dense matter.Rather,we would like to point out here that,if this decoupling happens in a certain temperature region(like it was quoted in[2]),then a pion condensate should appear.

If a pion condensate is created in a heavy ion collision,it may be distinguishable from the case of a resonance gas in thermal and chemical equilibrium essentially by the following e?ects.

?It should lead to a coherent component which then would appear in the two-particle correlation functions of identical pions through a reduction of the intercept of the correlation function and the appearance of a two exponent behaviour of the correla-tion function[13].

?The condensate component is moving with the?uid velocity and has no additional thermal component which smears out the distribution.This might lead to character-

istic bumps and shoulders in the rapidity and transverse momentum distributions of pions2.One also can expect that the coherent component disappears if one considers particles of velocities that exceed the maximum?uid velocities.

?The lifetime of the?reball may be reduced.This could at least partially compensate the enhancement of the lifetime expected[15]in the case that quark-gluon-plasma is formed in the intial stage.

Instructive and helpful discussions with R.M.Weiner,F.Navarra and H.Leutwyler are gratefully acknowledged.This work was supported in part by the Deutsche Forschungsge-meinschaft(DFG),the Federal Minister for Research and Technology(BMFT)under the contract no.06MR731,the Gesellschaft f¨u r Schwerionenforschung(GSI)and the Los Alamos National Laboratory.

References

[1]Proceedings of Quark Matter91,eds.T.C.Awes,F.E.Obenshain,F.Plasil,M.R.

Strayer and C.Y.Wong,Nucl.Phys.A544

(1990)513.

[3]S.Gavin and P.V.Ruuskanen,Phys.Lett.B262

(1992)1403.

[5]A.Mann and R.M.Weiner,Nouvo Cim.10A(1972)383;

S.Eliezer and R.M.Weiner,Phys.Rev D13

(1991)503.

[8]J.Sollfrank,P.Koch and U.Heinz,Phys.Lett.B252

,(1982)140.

[10]F.Cooper and G.Frye,Phys.Rev.D10

(1989)3735.

[12]U.Ornik,F.W.Pottag and R.M.Weiner,Phys.Rev.Lett.63

(1989)278;B242

(1990)181.

[15]G.Bertsch,M.Gong,L.McLerran,V.Ruuskanen and E.Sarkkinen,Phys.Rev.D37

(1988)1896;G.

Bertsch,Nucl.Phys.A498

FIGURE CAPTIONS

Figure1:

Left column:Dependence of the chemical potentials for baryons(dashed), strangeness(dotted)and pions(solid)on the chemical freeze-out temperature T ch.f.,for three di?erent values of the baryon number density b.Right column:The thermal(solid) and condensate(dashed)components of the pion number densities at chemical freeze-out.

Figure3:

如何看懂电路图(超级完整版)

如何看懂电路图1－－学电子跟我来系列文章 电子设备中有各种各样的图。能够说明它们工作原理的是电原理图，简称电路图。 电路图有两种，一种是说明模拟电子电路工作原理的。它用各种图形符号表示电阻器、电容器、开关、晶体管等实物，用线条把元器件和单元电路按工作原理的关系连接起来。这种图长期以来就一直被叫做电路图。 另一种是说明数字电子电路工作原理的。它用各种图形符号表示门、触发器和各种逻辑部件，用线条把它们按逻辑关系连接起来，它是用来说明各个逻辑单元之间的逻辑关系和整机的逻辑功能的。为了和模拟电路的电路图区别开来，就把这种图叫做逻辑电路图，简称逻辑图。 除了这两种图外，常用的还有方框图。它用一个框表示电路的一部分，它能简洁明了地说明电路各部分的关系和整机的工作原理。 一张电路图就好象是一篇文章，各种单元电路就好比是句子，而各种元器件就是组成句子的单词。所以要想看懂电路图，还得从认识单词——元器件开始。有关电阻器、电容器、电感线圈、晶体管等元器件的用途、类别、使用方法等内容可以点击本文相关文章下的各个链接，本文只把电路图中常出现的各种符号重述一遍，希望初学者熟悉它们，并记住不忘。 电阻器与电位器 符号详见图 1 所示，其中（ a ）表示一般的阻值固定的电阻器，（ b ）表示半可调或微调电阻器；（ c ）表示电位器；（ d ）表示带开关的电位器。电阻器的文字符号是“ R ”，电位器是“ RP ”，即在 R 的后面再加一个说明它有调节功能的字符“ P ”。 在某些电路中，对电阻器的功率有一定要求，可分别用图 1 中（ e ）、（ f ）、

（ g ）、（ h ）所示符号来表示。 几种特殊电阻器的符号： 第 1 种是热敏电阻符号，热敏电阻器的电阻值是随外界温度而变化的。有的是负温度系数的，用NTC来表示；有的是正温度系数的，用PTC来表示。它的符号见图（ i ），用θ或t° 来表示温度。它的文字符号是“ RT ”。 第 2 种是光敏电阻器符号，见图 1 （ j ），有两个斜向的箭头表示光线。它的文字符号是“ RL ”。 第 3 种是压敏电阻器的符号。压敏电阻阻值是随电阻器两端所加的电压而变化的。符号见图 1 （ k ），用字符 U 表示电压。它的文字符号是“ RV ”。这三种电阻器实际上都是半导体器件，但习惯上我们仍把它们当作电阻器。 第 4 种特殊电阻器符号是表示新近出现的保险电阻，它兼有电阻器和熔丝的作用。当温度超过500℃ 时，电阻层迅速剥落熔断，把电路切断，能起到保护电路的作用。它的电阻值很小，目前在彩电中用得很多。它的图形符号见图 1 （ 1 ），文字符号是“ R F ”。 电容器的符号 详见图2 所示，其中（ a ）表示容量固定的电容器，（ b ）表示有极性电容器，例如各种电解电容器，（ c ）表示容量可调的可变电容器。（ d ）表示微调电容器，（ e ）表示一个双连可变电容器。电容器的文字符号是 C 。 电感器与变压器的符号 电感线圈在电路图中的图形符号见图 3 。其中（ a ）是电感线圈的一般符号，（ b ）是带磁芯或铁芯的线圈，（ c ）是铁芯有间隙的线圈，（ d ）是带可调磁芯的可调电感，（ e ）是有多个抽头的电感线圈。电感线圈的文字符号是“ L ”。

The way常见用法

The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法） 举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法： 在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because

Origin 使用问题

Origin 使用问题集锦 1. 请教怎样反读出 origin 曲线上全部数据点？ 如，我用 10个数据点画出了一条 origin 曲线，并存为 project的.OPJ 格式。但，现在我想利用 OPJ 文件从这条曲线上均匀的取出 100个数据点的数值，该如何做？注：要一切都使用 origin 软件完成，不用其他曲线识别软件。 Answer： ORIGIN 中，在分析菜单（或统计菜单）中有插值命令，打开设置对话框，输入数据的起点和终点以及插值点的个数，OK！生成新的插值曲线和对应的数据表格。 2. 如何用origin 做出附件中的图： 其中标注的三角形、方块是怎么整上去的？ Answer: 选中左侧竖工具条中的 draw tool(显示是几个点，第七个工具），移动到你要标注的位置双击，就产生了一个点，依次标注完方块。再标注三角的第一个点，标注完后改成三角，以后标注的就都是三角了。改动点的类型的方法和正常画曲线方式一样。 3. 如何用origin 做出附件图中的坐标轴（带刻度）？

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初中语法祈使句专项讲解

初中语法祈使句专项讲解 1、祈使句的概念表示请求、命令、建议、祝愿、邀请或要求的句子叫祈使句。用于祈使句句首的动词总是用原形，不能用其他形式。如：Shut the door! 把门关上!Have a cup of coffee! 喝杯咖啡吧!Let them go by train、叫他们坐火车去吧。祈使句的主语通常为第二人称(you)，但一般都被省略，只有在特殊的情况下才把主语(you)补充出来。如：You be quiet! 你安静! You go and tell him, Chris、克立斯，你去告诉他。You wait here for a moment、你在这儿等一会儿。有时祈使句的主语也可以是everybody, somebody, anybody, nobody等不定代词。如：Stand up, everybody! 全体起立!Nobody move、任何人都不许动。 2、祈使句表示强调为了加强祈使句的语气，我们通常是在祈使句的动词原形前加上助动词do，此时通常译为“一定”“务必”等。如：Do be careful、务必要小心。Do let me go、一定让我去。副词never和always有时可用于祈使句句首，表示强调。如：Never do that again、再不要这样做了。Always look in the mirror before starting to drive、一定要先看看反光镜再开车。 3、祈使句的否定式构成祈使句否定式的方法很简单，那就是在动词原形前加don’t不管祈使句所用的动词为什么性质动

词，情况都是一样。如：Open the window、把窗户打开。 →Don’t open the window、别把窗户打开。Come next Monday、下周星期一来。对于以let us或let’s开头的祈使句，其否定式通常是在不定式之前放一个not。如：Let’s tell him the truth、我们把实情告诉他吧。→Let’s not tell him the truth、我们不要把实情告诉他。但在日常口语中，我们也可以把don’t放在let’s之前。如：Let’s don’t tell him the truth、我们不要把实情告诉他) 4、祈使句与连用please为了使祈使句的语气变得委婉，我们可以在祈使句的句首或句末加上please若加在句首，其后不用逗号；若加在句末，则通常会在please前加一个逗号。如：Step this way, please、请这边走。Please type your letter、请把你的信打出来。Open the window, please、请把窗户打开。如果是否定祈使句，则通常将please加在don’t之前。如：Please don’t get angry、请不要生气。Please don’t telephone before8 a、 m、早8点以前请不要打电话。 5、祈使句的时间概念祈使句所表示的时间总是指将来，所以与它连用的句子原则上要用将来时态来与它呼应。如：Give me a hand, will you? 帮我一个忙，好吗?由于give me a hand表示隐含将来意义，所以句中的will you 不能改为do you。Try again and you will succeed、你再试就会成功。由于句中的 try again隐含将来意义，所以句中用了you will succeed，而

The way的用法及其含义(二)

The way的用法及其含义（二） 二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语： 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势，忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中

origin8使用小技巧

人人网- 日志分享 1. 请教怎样反读出origin曲线上全部数据点？ 如，我用10个数据点画出了一条origin曲线，并存为project的.OPJ格式。 但，现在我想利用OPJ文件从这条曲线上均匀的取出100个数据点的数值，该如何做？ 注：要一切都使用origin软件完成，不用其他曲线识别软件。 https://www.wendangku.net/doc/0117069567.html,/bbs/viewthread.php?tid=1390313 [1] Answer： ORIGIN中，在分析菜单（或统计菜单）中有插值命令，打开设置对话框，输入数据的起点和终点以及插值点的个数，OK！生成新的插值曲线和对应的数据表格。 2. origin中非线性拟合中logistic模型的疑问？ origin 中非线性拟合中的logistic模型为 y = A2 + (A1-A2)/(1 + (x/x0)^p) 其初始参数设置为 sort(x_y_curve); //smooth(x_y_curve, 2); x0 = xaty50( x_y_curve ); p = 3.0;

A1 = max( y_data ); A2 = min( y_data ); A1 = min( y_data ); A2 = max( y_data ); 而据我看到的logistic的模型都是（自己origin中自定义的） y =A1/(1+(A1/A2-1)*exp(-k*x)) 也就是说 origin 中的logistic有4个数值需要确定，而自定义的有3个数值 从结果来看，没有太大区别，但为什么函数不一样呢？ 不是学数学，高人能否详细说明下。 https://www.wendangku.net/doc/0117069567.html,/bbs/viewthread.php?tid=1391522 [2] Answer: 你可以看一下这个文档，里面有数种不同形式的 logistic 模型： https://www.wendangku.net/doc/0117069567.html,/web/packages/drc/drc.pdf [3] 当然，这是一个 R (https://www.wendangku.net/doc/0117069567.html,) 包的文档，但不妨碍你看其中的公式。 R 是开源的啊，以 GPL 发布，可以从 https://www.wendangku.net/doc/0117069567.html, [4]上了解更多。I 3. 如何用origin做出附件中的图：其中标注的三角形、方块是怎么整上去的？https://www.wendangku.net/doc/0117069567.html,/bbs/viewthread.php?tid=1393739 [5] Answer:

祈使句用法

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祈使句的否定形式一般是在谓语动词前加上do not或don't（口语中）， 有时也可用never。若祈使句有主语，否定词don't或never要置于主语之前。 Do not come in uni ess asked.非请莫入。 Don't you believe i t .决不要相信它。 Don't anyone make any no ise .谁也不要吵吵闹闹。 Never be iate again next time. 下一次千万不要再迟到了。 4.祈使句的否定：iet let在祈使句中的否定形式有以下三种: 一、Don't iet Do n' t let the baby fall不要让婴儿摔下来。 Don' t let yourself be eate n by an amateU别让你自己别业余的人吃 掉。 二、Let …not Let ' s not think about it. I am tired of think about it. 咱们别再想那件事了。我已经厌倦了。 Let us not talk about that matter. 咱们别再谈论那个问题了。 三、Let…no （少用） Let there be no mistake about it. 以后别再犯这种错误了。 5.祈使句:动词原形多数的祈使句是以动词原形开头的

(完整版)the的用法

定冠词the的用法： 定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸 或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...

祈使句语法

祈使句的常见用法 几种形式： 1. 肯定祈使句（谓语动词用动词原形）。 Be careful!小心！ 2. 否定祈使句（在谓语动词前加Do not / Don't / Never）。 Don't open the door. 不要开门。 3. Let开头的祈使句（Let's / Let sb +动词原形）。 Let me have a try. 让我试试吧。 4. “No +名词或动名词”也可以构成祈使句。用于“指示标牌、布告”等，意为“禁止、不许”。 No spitting!不准随地吐痰！ No admittance except on business! 非公莫入！ 5. 无动词祈使句。有些用于口号、告示的祈使句不用动词，使语言更加简洁明了。如： After you! 您先请！ Bottoms up! 干杯！ 注意： 1. 祈使句一般不出现主语you。但是，为了加强语气或要特别指明向谁提出命令或要求时，需加主语you，有时还可以同时加称呼语。如：Tom, you water the flowers! 汤姆，你浇花！ Come in, everybody! 大家都进来！ 2. 祈使句后也可加简略问句，不表示反义，而是表示语气。如： Show me how to use this water pump, won't you? 教我怎样使用这台抽水泵，好吗？Let's make sure the lights are turned off and the door is locked, shall / shan't we? 让我们核实一下是否关灯和锁门了，好吗？ Let me tidy the lab instead of you, may I? 我替你收拾实验室，好吗？ 3. 祈使句也可用于表示祝愿。如： Wish you a good journey. 祝你旅途愉快。 Allow me to propose a toast to our friendship!为我们的友谊干杯！

“the way+从句”结构的意义及用法

“tｈｅwａｙ+从句”结构的意义及用法 首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。 下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句 例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法

不对。 Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.

常用电路图及电器的文字符号和图形符号

一、常用电路图- 1 -1．单按钮控制两台电动机顺序启动反序停止- 1 - 3．用两个时间继电器控制电动机间歇正反转- 2 -4．三地控制三相电动机正反转- 3 -5．两地控制一台电动机- 4 -6．频敏变阻启动原理图- 4 - 7．用一个时间继电器,和三个按钮,控制一个灯220和电机380,要求电机能自动运行60秒停止- 5 - 8. 接近开关导通后电机停止接近开关断开后延时N秒电机启动- 5 - 9.运用时间继电器使电磁铁动作2秒后复位,经过3分钟后动作2秒后复位,再经过5分钟后动作2秒复位- 6 - 10. 利用电接点压力表自动控制水泵- 6 - 11. 两台电动机既可分别启动和停止,也可以同时启动和停止. - 7 - 12. 正转停止后,必须过预定的时间(如5S)后才能反转,反转停止后,必须过预定的时间(如5S)后才能正转- 7 - 13. 用三个时间继电器控制正反转并要有间隙- 8 - 14. 三相异步电动机转子串联电阻启动- 8 -

15. 三相异步电动机启动控制线路图（带故障指示灯）- 9 - 16. 双控及多地控制（照明) - 10 - 18. 使电机有点动还有正常运行- 11 - 19. 用3个继电器控制电动机断相保护- 11 - 20. 用四个时间继电器控制正反转并要有间隙- 12 - 21. 三相电动机在220V电压下正反转能耗制动- 12 - 22. 三个地方控制一盏灯- 13 - 23. 星三角降压的电路用4个交流接触器和一个时间继电器要做成可以正反转的电路并且可以自动和手动的- 13 - 24. 延边三角形降压启动的原理图- 14 - 25. 点动与长动的正反转控制电路- 14 - 26. 用按钮开关(常开）启动电动机，用行程开关(常闭）停止电动机实物接线图- 15 -27用按钮开关(常开）启动电动机，用行程开关(常开）停止电动机实物接线图- 15 -28．四个地方控制一盏灯- 16 -29. 单相电能表加装互感器- 16 -31. 用一个3a的按钮通过继电器控制一个12v15a的电机- 17 -

祈使句的用法讲解

祈使句的用法讲解 1、概述 祈使句(imperative sentence)表命令、警告、提醒、建议、请求、叮嘱、号召等，谓语动词用 原形。主语you通常省略，句末用叹号或句号。 Knock at the door before entering, please.进来时请敲门！（建议） it.尝点这种果汁，也许你会要。（建议） Try some of this juice---perhaps you’ll take Look out! There is a train coming.注意！火车来了！（提醒） Wear strong shoes as we shall do a lot of walking. 请穿结实的鞋子，因为我们要走许多的路。（叮嘱） At the beginning, collect as many stamps as you can. 开始时，尽可能多多地收集邮票。（建议）Leave her where she is ! 让她留在原地！（命令） Put your coat at once. We must hurry.立刻穿上衣服。我们必须快点。（命令） Give blood if you can and many lives will be saved.如果你能，来献血吧，这会挽救很多人的 生命。（号召） 2、祈使句的另外表达方式 （1）“No＋名词或动名词”　表示祈使句 “No＋名词或动名词”结构表示禁止的祈使，一般用于指示、标牌、布告等。 No smoking! 禁止吸烟！No spitting!不准随地吐痰！ No litter! 不准乱扔果皮纸屑！No entry!不许入内！ No thoroughfare!禁止通行！No scribbling on the wall!墙上不准涂写！ No admittance except on business!非公莫入！ （2）“have d one…”结构表示祈使句 这种结构相当于stop doing，意为“禁止”。有时表示“结束、停止”。 Have done scolding him. 不要再责备他了。 Have done!住手！ Have done running!跑完了！（别跑了！） （3）“be＋V-ed”结构表示祈使句 Be guided by reason!理智些！Be seated! 请坐！ Be persuaded by your father! 听你父亲的话！Be gone! 滚！ （4）“be+ V-ing”　 结构表示祈使句 be looking out of the window! 勿将头伸出窗外！ Don’ｔ Do be doing your homework when I come in.我进来时，你一定在做作业吧！ （5）无动词祈使句 一般来讲，祈使句都要有动词，但是有些用于口号、告示等的祈使句却是没有动词。主要 有：名词短语；介词短语；名词＋代词／副词／形容词；形容词＋名词／代词；副词＋with 等。 Don’t all speak at once! O ne at a time, please.不要一起说，一次一个！ 再给我一星期，我就会完成任务。 One more week, and we’ll accomplish the task. Patience! 耐心点！Your turn. 轮到您了。After you! 您先请！ Now for it! 干起来吧！On with your cap. 带上帽子。Danger!危险！ Poison!有毒！Over! 完毕！Bottom up!干杯！ None of that again! 不要在那样！None of your nonsense!不要胡说八道！ Hand up!举起手来！Hands off!无动手！ Off with it!把它拿下来！Up with the box!把箱子放下！

way 用法

表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。 如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如： 我不喜欢他说话的态度。 正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。 一、in a way的用法 表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。 二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。 三、by the way的用法

祈使句用法

祈使句 1、概述 祈使句(imperative sentence)表命令、警告、提醒、建议、请求、叮嘱、号召等，谓语动词用原形。主语you通常省略，句末用叹号或句号。 Knock at the door before entering, please.进来时请敲门！（建议） Try some of this juice---perhaps you’ll take it.尝点这种果汁，也许你会要。（建议） Look out! There is a train coming.注意！火车来了！（提醒） Wear strong shoes as we shall do a lot of walking. 请穿结实的鞋子，因为我们要走许多的路。（叮嘱） At the beginning, collect as many stamps as you can. 开始时，尽可能多多地收集邮票。（建议）Leave her where she is ! 让她留在原地！（命令） Put your coat at once. We must hurry.立刻穿上衣服。我们必须快点。（命令） Give blood if you can and many lives will be saved.如果你能，来献血吧，这会挽救很多人的生命。（号召） 2、祈使句的另外表达方式 （1）“No＋名词或动名词”表示祈使句 “No＋名词或动名词”结构表示禁止的祈使，一般用于指示、标牌、布告等。 No smoking! 禁止吸烟！No spitting!不准随地吐痰！ No litter! 不准乱扔果皮纸屑！No entry!不许入内！ No thoroughfare!禁止通行！No scribbling on the wall!墙上不准涂写！ No admittance except on business!非公莫入！ （2）“have done…”结构表示祈使句 这种结构相当于stop doing，意为“禁止”。有时表示“结束、停止”。 Have done scolding him. 不要再责备他了。 Have done!住手！ Have done running!跑完了！（别跑了！） （3）“be＋V-ed”结构表示祈使句 Be guided by reason!理智些！Be seated! 请坐！ Be persuaded by your father! 听你父亲的话！Be gone! 滚！ （4）“be+ V-ing”结构表示祈使句 Don’t be looking out of the window! 勿将头伸出窗外！ Do be doing your homework when I come in.我进来时，你一定在做作业吧！ （5）无动词祈使句 一般来讲，祈使句都要有动词，但是有些用于口号、告示等的祈使句却是没有动词。主要有：名词短语；介词短语；名词＋代词／副词／形容词；形容词＋名词／代词；副词＋with等。Don’t all speak at once! One at a time, please.不要一起说，一次一个！ One more week, and we’ll accomplish the task.再给我一星期，我就会完成任务。 Patience! 耐心点！Your turn. 轮到您了。After you! 您先请！ Now for it! 干起来吧！On with your cap. 带上帽子。Danger!危险！ Poison!有毒！Over! 完毕！Bottom up!干杯！ None of that again! 不要在那样！None of your nonsense!不要胡说八道！ Hand up!举起手来！Hands off!无动手！ Off with it!把它拿下来！Up with the box!把箱子放下！

The way的用法及其含义(一)

The way的用法及其含义（一） 有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析： 一、the way作先行词，后接定语从句 以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.

Origin 使用问题集锦

1. 请教怎样反读出origin曲线上全部数据点？ 如，我用10个数据点画出了一条origin曲线，并存为project的.OPJ格式。 但，现在我想利用OPJ文件从这条曲线上均匀的取出100个数据点的数值，该如何做？注：要一切都使用origin软件完成，不用其他曲线识别软件。 https://www.wendangku.net/doc/0117069567.html,/bbs/viewthread.php?tid=1390313 Answer： ORIGIN中，在分析菜单（或统计菜单）中有插值命令，打开设置对话框，输入数据的起点和终点以及插值点的个数，OK！生成新的插值曲线和对应的数据表格。 2. origin中非线性拟合中logistic模型的疑问？ origin 中非线性拟合中的logistic模型为 y = A2 + (A1-A2)/(1 + (x/x0)^p) 其初始参数设置为 sort(x_y_curve); //smooth(x_y_curve, 2); x0 = xaty50( x_y_curve ); p = 3.0; if( yatxmin( x_y_curve ) > yatxmax( x_y_curve ) ) { A1 = max( y_data ); A2 = min( y_data ); } else { A1 = min( y_data ); A2 = max( y_data ); } 而据我看到的logistic的模型都是（自己origin中自定义的） y =A1/(1+(A1/A2-1)*exp(-k*x)) 也就是说origin 中的logistic有4个数值需要确定，而自定义的有3个数值 从结果来看，没有太大区别，但为什么函数不一样呢？ 不是学数学，高人能否详细说明下。 https://www.wendangku.net/doc/0117069567.html,/bbs/viewthread.php?tid=1391522 Answer: 你可以看一下这个文档，里面有数种不同形式的logistic 模型： https://www.wendangku.net/doc/0117069567.html,/web/packages/drc/drc.pdf 当然，这是一个R (https://www.wendangku.net/doc/0117069567.html,) 包的文档，但不妨碍你看其中的公式。 R 是开源的啊，以GPL 发布，可以从https://www.wendangku.net/doc/0117069567.html,上了解更多。