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Squeeze-out of nuclear matter in peripheral heavy-ion collisions and momentum-dependent eff

Squeeze-out of nuclear matter in peripheral heavy-ion collisions and momentum-dependent eff
Squeeze-out of nuclear matter in peripheral heavy-ion collisions and momentum-dependent eff

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Squeeze-out of nuclear matter in peripheral heavy-ion collisions and momentum-dependent e?ective interactions.?https://www.doczj.com/doc/0a5948687.html,rionov ?,W.Cassing,C.Greiner,U.Mosel Institut f¨u r Theoretische Physik,Universit¨a t Giessen,D-35392Giessen,Germany Abstract We perform a systematic study of in-plane and out-of-plane proton and neutron ?ow from nucleus-nucleus collisions within the BUU transport ap-proach employing di?erent parameter sets for the mean-?eld potentials that can be characterized by the nuclear incompressibility and the sti?ness of the momentum-dependent potential.We ?nd that a simultaneous description of the experimental data from the BEVALAC and the SIS on both the nucleon squeeze-out v 2and the in-plane ?ow F at beam energies E lab =0.15÷2AGeV requires a mean ?eld with strong momentum dependence,i.e.an e?ec-tive Landau mass m ??0.68m 0at normal nuclear matter density ρ0=0.17fm ?3,where m 0=0.938GeV is the bare nucleon mass.Some experimen-tal data on the squeeze-out require an even sti?er momentum dependence

(m ?≈0.62m 0).All systems investigated are found to be compatible with

m ?/m 0=0.65±0.03.

PACS numbers:25.70.-z;25.75.-q;25.75.Ld;21.65.+f

Keywords :Heavy-ion collisions;Collective ?ow;Transport models;Equation of state;E?ective mass

I.INTRODUCTION

Since many years the study of the various collective?ows of nuclear matter is one of the main subjects in heavy-ion physics[1–9].The interest in collective nuclear motion under extreme conditions,like high density and/or high temperature,originates from the fundamental problem to extract the equation of state(EOS)of nuclear matter.It was shown in Refs.[10–13]that the in-plane directed?ow[14]is sensitive to both the incompressibility K of the EOS and to the momentum-dependent e?ective interaction(MDI).Moreover,the in-medium nucleon-nucleon(NN)cross section(that is related to the imaginary part of the in-medium G-matrix)also in?uences the transverse directed?ow[15]since nucleons acquire transverse momentum also by elastic and inelastic collisions.It is necessary,therefore,to have at least two additional observables,besides the in-plane?ow,in order to determine the three most commonly used components of the e?ective NN interaction adopted in the various analyses:the incompressibility K,the sti?ness of the MDI and,possibly,the in-medium NN cross section.

The rapidity distribution of nucleons,which is only weakly dependent on the mean-?eld parameters,gives reasonable constraints on the in-medium NN cross section or on the nuclear stopping power[16,17].However,it was not so clear until recently,how to possibly extract information on the MDI from heavy-ion collisions.Danielewicz[18]has shown within the Boltzmann-Uehling-Uhlenbeck(BUU)transport model,that the elliptic ?ow(or squeeze-out ratio)of protons at midrapidity is very sensitive to the MDI part of the nucleon mean?eld.The reaction Bi+Bi at E lab=400,700,1000AMeV(measured by the KAOS Collaboration[19])has been analysed in[18]leading to the conclusion that the MDI part of the NN interaction should produce an e?ective Landau mass m??0.7m0,since momentum independent forces(m?=m0)strongly underpredict the squeeze-out ratio at high transverse momenta[18].

The aim of our present work is:(i)to study the mechanism of the squeeze-out enhance-ment by the MDI and(ii)to analyse several independent data sets on the directed and elliptic

?ow of nucleons to get experimental constraints on the sti?ness of the MDI.We concentrate on the bombarding energy regime up to2AGeV since here multiple meson production or ”string”degrees of freedom play a minor role.As shown by Sahu et al.[20]the collective?ow is strongly in?uenced by the e?ective degrees of freedom taken into account in the transport approach at high energies.For a detailed investigation of this problem–especially at AGS energies–we refer the reader to Ref.[20].

The paper is structured as follows:In Sect.II a brief description of the applied BUU model is given.Sect.III contains our interpretation of the squeeze-out mechanism which is illustrated in Sect.IV by a detailed study of the time evolution of colliding nuclear systems within BUU calculations.A comparison to the experimental data on the proton and neutron elliptic and directed?ows is given in Sect.V,while Sect.VI concludes our work with a summary of the results.

II.THE BUU MODEL

In our calculations we have applied the recent version of the BUU model described in more detail in Ref.[21].The equation of motion for the nucleon phase-space distribution function f(r,p,t)includes a propagation in the momentum-dependent scalar mean?eld s(r,p,f)via the single-particle energy

H mf(r,p,t)=

?t +

?H mf

?r

?

?(H mf+U Coul)

?p1 f(r,p1,t)

=g d p2d?(f3f4(1?f1)(1?f2)?f1f2(1?f3)(1?f4))

+coupling terms,(2) where U Coul is the Coulomb potential acting on protons;g=4is the spin-isospin degeneracy of a nucleon;v12=|v1?v2|and p12=|p1?p2|are the relative velocity and relative momen-

tum of colliding nucleons,respectively;dσNN(p12)/d?is the energy dependent di?erential NN scattering cross section as parametrized by Cugnon[22];f i:=f(r,p i,t)(i=1,2,3,4), while p3and p4are the?nal momenta of the scattered nucleons.The coupling terms in(2) account for the various inelastic channels:NN?πNN,NN?NR–S-wave pion and resonance production/absorption,NN→ΛKN,NN→Σ0KN–strangeness production.

The scalar potential s(r,p,f)is computed as follows(see[23]):

1)In the local rest frame(l.r.f.)of the nuclear matter element,where the space components of the baryonic4-current jμ=(j0,j)vanish,i.e.j=0,the single-particle energy is calculated as

?l.r.f.(r,p)=

+B ρρ0 gd p′1+(p?p′)2/Λ2.(4)

ρ0

A symmetry energy term is not taken into account in the potential(4).

2)The mean?eld s is calculated from the relation in the l.r.f.

s=

π2ρ)1/3ˉh.The residual four free parameters A,B,τand C of the

2

potential U are determined from the conditions:

(i)The e?ective mass at the Fermi surface

?U

(m?)?1=m?10+(p(0)F)?1

must be in the range0.6m0≤m?≤m0(c.f.

[25,26]).This

condition puts limits on the sti?ness C of the MDI.

(ii)The energy per nucleon in nuclear matter has a minimum atρ=ρ0and

(iii)a value of?16MeV at the minimum:

??/ρ

ρ|ρ=ρ

=?16MeV,(7) where

?(ρ)=

3

2

A

ρ2

τ+1

B

ρτ+1

ρ0 gd p1(2πˉh)3f(p1)f(p2)

?ρ2|ρ=ρ

(9) is?tted in the range K=200÷380MeV.

We have applied in our calculations the following mean?elds denoted by H(hard,with-out momentum dependence),HM and SM(hard and soft,with a”standard”momentum dependence)as well as the new mean-?eld parametrization which is dubbed HM1(see be-low).The parameters of all interactions are presented in Table1.The sti?ness C of the MDI in the interactions HM and SM is close to that of Ref.[24].The set of parameters HM1includes an MDI sti?ness that is about30%larger than in HM and SM,which leads to a smaller e?ective mass at the Fermi momentum for HM1.In Fig.1we show the resulting energy per nucleon as a function of density and the potential U(ρ,p)versus momentum for several densities.The interactions H,HM and HM1give practically the same EOS(E/A(ρ)-dependence),but they di?er in the MDI part,i.e.in their momentum dependence.This gives the possibility to extract an independent information on the MDI sti?ness C.On the other hand,the interactions HM and SM produce di?erent EOS,but have the same MDI, which is relevant for a separate study on the sensitivity to the incompressibility K.

III.SQUEEZE-OUT MECHANISM

A squeeze-out of nucleons is observed experimentally in the beam energy range below1÷2

AGeV where mean-?eld e?ects still play an essential role.It can be interpreted,however, to a major extent in terms of a shadowing phenomenon[5],as in the participant-spectator picture of a heavy-ion collision the?reball particles–emitted in the reaction plane–are rescattered on the spectators.This results intuitively in a squeeze-out ratio R N:=(N(90o)+ N(270o))/(N(0o)+N(180o))at midrapidity being larger than1.Here N(φ)is the azimuthal distribution of nucleons with respect to the reaction plane in a given rapidity interval.

A speci?c particle emitted from the center-of-mass of a system and moving in transverse direction with velocity v t will be shadowed by the spectator piece,if it reaches the spectator during the passage time?t=2R/(v c.m.

p

γ)of the colliding nuclei.Here R is the radius of the

nuclei(assummed to be equal for projectile and target),v c.m.

p

is the projectile velocity in the center-of-mass system,andγ=(

R γv c.m.

p

,(10)

where b is the impact parameter.One can see from(10)that at larger impact parameters b the lower limit of v t becomes smaller,i.e.more particles will be shadowed and the squeeze-out ratio R N will increase(https://www.doczj.com/doc/0a5948687.html,ter Fig.12).This is simply related to a larger size(~b) of the spectator pieces in peripheral collisions.It is also evident,that for faster moving particles it is easier to ful?ll the condition(10)and,therefore,the squeeze-out ratio should grow with the particle transverse velocity v t(https://www.doczj.com/doc/0a5948687.html,ter Figs.6,13).

The squeeze-out phenomenon can also be characterized by a negative elliptic?ow v2(c.f.

[27])which appears in the Fourier expansion

N(φ)∝1+2v1cos(φ)+2v2cos(2φ)+ (11)

Neglecting thus higher order terms in(11),the following direct relation is obtained:

R N ?1?2v 2

p 2x +p 2y .(13)

At higher beam energy (~10AGeV),faster moving and Lorentz-contracted spectators do not shadow particles any more from the expanding ?reball

[20].This results in R N ≤1(or v 2≥0)as observed in experiment [28].

The simple geometrical picture discussed above will be illustrated by the BUU phase-space evolution in the next section.Since the squeeze-out is related to the participant-spectator reaction mechanism (c.f.[29]),we expect a lower beam-energy limit of ~0.1AGeV for R N >1.At even lower beam energies the mean-?eld e?ects lead to the formation of rotating nuclear systems [30]that emit particles predominantly in the reaction plane due to centrifugal forces,thus producing R N <1(or v 2>0,https://www.doczj.com/doc/0a5948687.html,ter Fig.7).

We have to note,however,that the shadowing scenario can not completely explain the squeeze-out phenomenon.For instance,a small but statistically signi?cant increase of ?v 2at the beam energy ~0.4AGeV (see later Fig.7)with increasing incompressibility K can not be understood in the shadowing picture.This is,probably,a manifestation of a focussing of high momentum particles by the repulsive mean ?eld as proposed in Ref.[18].

IV.BUU STUDY OF THE SQUEEZE-OUT TIME EVOLUTION

We study the system Au+Au at the beam energy E lab =0.4AGeV and impact parameter b =6fm.Fig.2shows the time evolution of the central baryon density and of the NN collision rate for the HM and SM mean ?elds.In agreement with Fig.1(upper left panel),a slightly higher density and,as a consequence,a higher collision rate are reached in the case of the SM parametrization.However,the various EOS shown in Fig.1practically do not di?er in the density interval 0<ρ≤0.27fm ?3probed in the collision.We do not expect,

therefore,to get clear constraints on the incompressibility K from our study of peripheral collisions and concentrate mainly on the MDI-part of the e?ective interactions.

As pointed out in Ref.[18]the p t-dependence of the v2coe?cient is very sensitive to the momentum dependence of the nuclear mean?eld.In order to understand this fact,we have performed calculations with the parametrizations H and HM of the mean?eld.Fig.3 shows the average transverse velocity(a)and transverse momentum(b)of the midrapidity neutrons moving in-and out-of the reaction plane versus time.We observe that in the case of the HM parameter set v t reaches a maximum at t?17fm/c when also the central density and the collision rate are close to the maximum(Fig.2).At this time the system shows the most compact con?guration in space,i.e.the line connecting the centers of target and projectile spectators,is perpendicular to the beam(z-)direction(Fig.4).However,in case of the H mean?eld the transverse velocity has no maximum(the peaks of v t and p t at t?7fm/c are due to?uctuations,because initially there are no nucleons at midrapidity). Moreover,at t<40fm/c in the case of the HM mean?eld the average transverse velocity is larger than for the mean?eld H,while the transverse momentum is practically independent on the mean?eld,even showing a slightly opposite tendency:at t<40fm/c the mean ?eld H produces a somewhat larger p t .This can be qualitatively understood from the Hamiltonian equations:

˙r=?H mf

H mf

+

m0+s

?p

,(14)

˙p=??H mf

H mf

?s

of a mean?eld.

Later on v t and p t decrease reaching asymptotic values at t?50fm/c.This decrease is due to less energetic products of NN scattering,which populate the midrapidity region at later times,while?rst-chance collision products have larger kinetic energies.Faster moving particles(in the case of the HM calculation with respect to H)are shadowed more e?ectively, which causes a larger splitting between in-and out-of-plane transverse velocities for the HM calculation at large times.

In Fig.5we show the?nal azimuthal angle dependence of the midrapidity neutron trans-verse velocity(a)and the neutron azimuthal distribution(b).The azimuthal angle modu-lation of the transverse velocity is clearly visible,which was?rst observed experimentally [31].The transverse velocity azimuthal angle dependence and the azimuthal distribution of particles have the same shape.The depletion of the particle yield atφ=0o and180o is caused by a shadowing of fast moving particles by the spectators;the enhanced yield at φ=±90o correlates with the nonshadowed emission of fast particles.

Fig.6shows the coe?cient v2vs.transverse velocity(left panels)and vs.transverse momentum(right panels)at the two time steps t=30fm/c and t=50fm/c.Even at30 fm/c the particles still feel the in?uence of the nuclear?eld and,therefore,the simple relation v t=p t/H mf can be applied at this moment only in the case of the momentum-independent mean?eld H.One can see,that at30fm/c the v2(p t)-dependence is di?erent for di?erent mean?elds,while the v2(v t)-dependence is very similar for the H and HM calculations. This is in agreement with our expectation,that particles moving with the same transverse velocity are shadowed in the same way regardless of their transverse momenta.At later times,this”transient universality”of v2(v t)is destroyed since the in?uence of the mean ?eld will become negligible here and v t=p t/

https://www.doczj.com/doc/0a5948687.html,PARISON WITH EXPERIMENT

In our comparison to data on squeeze-out observables we now compare in parallel also to data on the in-plane?ow F,which is de?ned as the derivative of the transverse in-plane momentum component p x with respect to the normalized c.m.rapidity y(0)=(y/y proj)c.m. at midrapidity[32]:

d p x

F=

in-plane?ow(Figs.8,9)increases monotonically in the beam energy region considered.The balance energy,i.e.the beam energy at which the?ow F changes sign,is E BAL?50AMeV in our calculation.This value is in between the MSU-data[38](E BAL?42AMeV)and the FOPI-data[39](E BAL?60AMeV).

Now we discuss the e?ect of the mean?eld on v2and F.The calculation with a momentum-independent mean?eld H is in a satisfactory agreement with proton in-plane?ow data below1AGeV(Fig.8),but it fails to reproduce the strong squeeze-out at E lab?0.4 AGeV(Fig.7).The HM parametrization increases the in-plane?ow and squeeze-out(see also Fig.5)and,thus,the in-plane proton?ow F is now overpredicted.The lower incom-pressibility K in the SM calculation results in a reduced squeeze-out and a reduced?ow with respect to the HM calculation.Generally,the in-plane proton?ow and v2data are best reproduced with the SM mean?eld.

In Fig.9we compare to FOPI,EOS and recent MSU-4πdata on the in-plane?ow excitation function.The selected data sets are now for a mixture of protons and complex fragments and,therefore,they exhibit a stronger?ow signal(c.f.Fig.8,where the data are for protons only).A separate study of the fragment?ow is out of scope of this work. However,in order to understand qualitatively the possible e?ects from cluster formation, we present in Fig.9the results of the BUU-SM calculation both for all(solid line)and free (dashed line)protons.We?nd a steeper increase of the in-plane?ow for all protons with the beam energy from0.1to1AGeV.The calculation for all protons is in a good agreement with data at E lab=0.4÷1AGeV,but it underpredicts the?ow at lower beam energies, where the free proton?ow F increases earlier with E lab and is closer to the data.This can be explained by the cluster formation scenario from initially free emitted protons by the momentum(rather than coordinate)space coalescence.The balance energy given by the SM calculation is E bal?50(60)AMeV for free(all)protons,which is somewhat larger than the value E bal?42AMeV reported in[38].But the shape of the in-plane?ow excitation function at E lab=25÷60AMeV is in a remarcable agreement with the MSU-4πdata.We would like to stress,that the negative in-plane?ow in the Au+Au system below E BAL is

obtained in our calculations taking into account initial repulsive Rutherford-trajectories(c.f. Ref.[34],where an opposite result has been obtained).

In Fig.10we present the midrapidity neutron azimuthal distributions with respect to the in-plane?ow axis for the reactions Nb+Nb and La+La at400AMeV in comparison to the data from Ref.[40](see Fig.11of[40]).In this comparison the BUU events were treated as the experimental data[40],i.e.the azimuthal angle of the reaction plane was de?ned by the Q vector:

Q= νwνvνt

vνi vνj

2m0

Finally,we have studied the neutron squeeze-out in Au+Au collisions measured by the FOPI-LAND Collaboration[44].Fig.12shows the impact parameter dependence of the squeeze-out ratio R N for Au+Au collisions at400AMeV,which reasonably describes the data for the HM calculation(dotted histogram).The HM1parameter set(solid histogram) underpredicts somewhat the squeeze-out ratio in central collisions and overpredicts it in semiperipheral(b=7÷8fm)collisions.The SM parameter set(dash-dotted histogram) and momentum-independent?eld H(dashed histogram)underpredict the data at large im-pact parameters.The calculated R N(b)-dependence has a characteristic bell-shape with a maximum at b?7fm.This agrees with the results of Ref.[39],where the squeeze-out was studied for charged particles.The drop of R N at large impact parameters can be explained by a smaller compression of the participant zone and,therefore,smaller transverse velocities of particles emitted in peripheral collisions.

In Fig.13the p t-dependence of the ratio R N is shown for collisions of Au+Au at400, 600and800AMeV.In agreement with the previous calculations of Ref.[18]we observe that a lower e?ective mass,i.e.a steeper momentum dependence,increases the squeeze-out ratio R N at large p t(c.f.dotted and solid lines in Fig.13d-f).On the other hand,reducing the NN cross section decreases the ratio R N at large p t(c.f.solid and long-dashed lines in Fig. 13e).The parameter set H produces a too?at R N(p t)-dependence(c.f.short-dashed line in Fig.13a).The SM calculation gives the best overall agreement with the data(dash-dotted line in Fig.13a-c).We see that our BUU calculations reproduce the experimentally observed trend of R N to decrease(at?xed p t)with collision energy,which was interpreted in Ref.[44]as a scaling behaviour of the squeeze-out.

VI.SUMMARY AND CONCLUSIONS

The squeeze-out of nuclear matter in heavy-ion collisions at beam energies E lab=0.05÷2 AGeV has been studied within the BUU approach of Ref.[21].As demonstrated,being essentially caused by the shadowing of the expanding?reball by cold spectators,the squeeze-

out is quite sensitive to the speci?c velocity of the particles.A mean?eld with a larger MDI sti?ness gives a smaller e?ective mass,and thus a harder velocity spectrum of nucleons. In a heavy-ion collision thus faster moving nucleons are shadowed more e?ectively and the squeeze-out ratio R N is larger for mean?elds with stronger momentum dependence.

The systematic comparison with the experimental data on the excitation function of the elliptic?ow v2[8]favours mean?elds with m??0.68m0,i.e.HM or SM sets.The in?uence of the incompressibility K on the squeeze-out turns out to be quite weak.The proton in-plane?ow data of the EOS Collaboration at E lab=0.25÷1.15AGeV and the fragment?ow data of the MSU-4πCollaboration at E lab=25÷60AMeV are consistent with the SM mean?eld.We have found,that a reproduction of the BERKELEY data[40]on the neutron azimuthal distributions and the in-plane?ow in Nb+Nb and La+La collisions at400AMeV requires an enhanced MDI sti?ness for p<1GeV/c(HM1parameter set: K=379MeV and m?=0.62m0).However,the FOPI-LAND data[44]on the neutron squeeze-out in Au+Au collisions at400÷800AMeV favour the standard MDI sti?ness(HM or SM parameter sets).

To conclude,a simultaneous description of the nucleon squeeze-out v2and the in-plane ?ow F at the beam energies E lab=0.2÷2AGeV requires a mean?eld with a strong momentum dependence.This corresponds to an e?ective Landau mass at the Fermi surface m?/m0=0.65±0.03at normal nuclear density.The incompressibility K,however,is less well determined since in semiperipheral reactions the average density probed up to1AGeV is too low.

ACKNOWLEDGMENTS

One of us(A.B.L.)is grateful to P.Danielewicz,W.G.Lynch,M.B.Tsang and G.D. Westfall for useful discussions during his stay at MSU,and to P.Danielewicz for pointing out the importance of a proper fragment selection in the comparison with fragment?ow data.

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R.Holzmann,J.G.Keller,H.Klingler,J.V.Kratz,R.Kulessa,https://www.doczj.com/doc/0a5948687.html,nge,Y.Leifels,E.

Lubkiewicz,E.F.Moore,W.Prokopowicz,R.Schmidt,C.Sch¨u tter,H.Spies,K.Stelzer, J.Stroth,E.Wajda,W.Walu′s,M.Zinser,E.Zude,and FOPI Collaboration,Z.Phys.

A350,115(1994).

Table1.Parameter sets for the di?erent mean?elds employed in the BUU calculation.

Not.m?/m0B(MeV)τ

H 1.0071.0 2.00

379-10.0-63.6 2.13

HM10.6229.5 2.62

220-108.6-63.6 2.13

FIGURE CAPTIONS

Fig.1Density dependence of the energy per nucleon in nuclear matter(upper left panel).

The other three panels show the mean-?eld potential vs.momentum at the nuclear densities0.5ρ0,ρ0and2ρ0.The dashed,dotted,solid and dash-dotted lines correspond to the interactions H,HM,HM1and SM,respectively.

Fig.2Central baryon density(smooth lines,left scale)and NN collision rate(histograms, right scale)vs.time for Au+Au at0.4AGeV and b=6fm.The solid(dashed)line and histogram correspond the HM(SM)calculation.The horizontal dotted line shows the normal nuclear densityρ0for comparison.

Fig.3Average transverse velocity(a)and transverse momentum(b)of neutrons in the c.m.

rapidity range|y|<0.1emitted in the reaction plane(φ=?30o÷30o,150o÷180o,?180o÷?150o)and out of the reaction plane(φ=60o÷120o,?60o÷?120o)for Au+Au at400AMeV and b=6fm.

Fig.4Time evolution of the baryon density in the reaction plane(x,z)for Au+Au at0.4 AGeV and b=6fm calculated with the HM mean?eld.Isolines from outer to inner correspond to densities in the order0.05,0.1,0.15,0.20and0.25fm?3.

Fig.5(a)Azimuthal dependence of the average neutron velocity and(b)neutron azimuthal distribution for Au+Au at0.4AGeV and b=5÷7fm.Neutrons are selected in the

c.m.rapidity interval|y|<0.1.The corresponding elliptic?ows are:v2=?0.046(H)

and v2=?0.090(HM).

Fig.6The elliptic?ow v2vs.transverse velocity(left panels)and vs.transverse momen-tum(right panels)of neutrons in the c.m.rapidity interval|y|<0.1for the time steps t=30fm/c(upper panels)and t=50fm/c(lower panels).The calculations have been performed with the H(open circles,dashed line)and HM(full circles,solid line) parameter sets.The system is Au+Au at0.4AGeV and b=6fm.

英语中的比较级与最高级 详解

比较级与最高级 1.as...as 与(not) as(so)...as as...as...句型中,as的词性 第一个as是副词,用在形容词和副词的原级前,常译为“同样地”。第二个as是连词,连接与前面句子结构相同的一个句子(相同部分常省略),可译为“同..... He is as tall as his brother is (tall) . (后面的as 为连词) 只有在否定句中,第一个as才可换为so 改错: He is so tall as his brother.(X) 2.在比较状语从句中,主句和从句的句式结构一般是相同的 与as...as 句式中第二个as一样,than 也是连词。as和than这两个连词后面的从句的结构与前面的句子大部分情况下结构是相同的,相同部分可以省略。 He picked more apples than she did. 完整的表达为: He picked more apples than she picked apples. 后而的picked apples和前面相同,用did 替代。 He walked as slowly as she did.完整表达为: He walked as slowly as she walked slowly. she后面walked slowly与前面相同,用did替代。

3.谓语的替代 在as和than 引导的比较状语从句中,由于句式同前面 主句相同,为避免重复,常把主句中出现而从句中又出现的动词用do的适当形式来代替。 John speaks German as fluently as Mary does. 4.前后的比较对象应一致 不管后面连词是than 还是as,前后的比较对象应一致。The weather of Beijing is colder than Guangzhou. x than前面比较对象是“天气”,than 后面比较对象是“广州”,不能相比较。应改为: The weather of Bejing is colder than that of Guangzhou. 再如: His handwriting is as good as me. 应改为: His handwriting is as good as mine. 5.可以修饰比较级的词 常用来修饰比较级的词或短语有: Much,even,far,a little,a lot,a bit,by far,rather,any,still,a great deal等。 by far的用法: 用于强调,意为“...得多”“最最...”“显然”等,可修饰形容词或副词的比较级和最高级,通常置于其后,但是若比较级或最高级前有冠词,则可置于其前或其后。

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

人教版(新目标)初中英语形容词与副词的比较级与最高级

人教版(新目标)初中英语形容词与副词的比较级与最高级 (一)规则变化: 1.绝大多数的单音节和少数双音节词,加词尾-er ,-est tall—taller—tallest 2.以不发音的e结尾的单音节词和少数以-le结尾的双音节词只加-r,-st nice—nicer—nicest , able—abler—ablest 3.以一个辅音字母结尾的重读闭音节词或少数双音节词,双写结尾的辅音字母,再加-er,-est big—bigger—biggest 4.以辅音字母加y结尾的双音节词,改y为i再加-er,-est easy—easier—easiest 5.少数以-er,-ow结尾的双音节词末尾加-er,-est clever—cleverer—cleverest, narrow—narrower—narrowest 6.其他双音节词和多音节词,在前面加more,most来构成比较级和最高级 easily—more easily—most easily (二)不规则变化 常见的有: good / well—better—best ; bad (ly)/ ill—worse—worst ; old—older/elder—oldest/eldest many / much—more—most ; little—less—least ; far—farther/further—farthest/furthest

用法: 1.原级比较:as + adj./adv. +as(否定为not so/as + adj./adv. +as)当as… as中间有名字时,采用as + adj. + a + n.或as + many / much + n. This is as good an example as the other is . I can carry as much paper as you can. 表示倍数的词或其他程度副词做修饰语时放在as的前面 This room is twice as big as that one. 倍数+as+adj.+as = 倍数+the +n.+of Your room is twice as larger as mine. = Your room is twice the size of mine. 2.比较级+ than 比较级前可加程度状语much, still, even, far, a lot, a little, three years. five times,20%等 He is three years older than I (am). 表示“(两个中)较……的那个”时,比较级前常加the(后面有名字时前面才能加冠词) He is the taller of the two brothers. / He is taller than his two brothers. Which is larger, Canada or Australia? / Which is the larger country, Canada or Australia? 可用比较级形式表示最高级概念,关键是要用或或否定词等把一事物(或人)与其他同类事物(或人)相分离 He is taller than any other boy / anybody else.

英语中的比较级和最高级

大多数形容词有三种形式,原级,比较级和最高级, 以表示形容词说明的性质在程度上的不同。 形容词的原级: 形容词的原级形式就是词典中出现的形容词的原形。例如: poor tall great glad bad 形容词的比较级和最高级: 形容词的比较级和最高级形式是在形容词的原级形式的基础上变化的。分为规则变化和不规则变化。 规则变化如下: 1) 单音节形容词的比较级和最高级形式是在词尾加 -er 和 -est 构成。 great (原级) (比较级) (最高级) 2) 以 -e 结尾的单音节形容词的比较级和最高级是在词尾加 -r 和 -st 构成。wide (原级) (比较级) (最高级) 3)少数以-y, -er, -ow, -ble结尾的双音节形容词的比较级和最高级是在词尾加 -er 和 -est 构成。 clever(原级) (比较级) (最高级) 4) 以 -y 结尾,但 -y 前是辅音字母的形容词的比较级和最高级是把 -y 去掉,加上 -ier 和-est 构成. happy (原形) (比较级) (最高级) 5) 以一个辅音字母结尾其前面的元音字母发短元音的形容词的比较级和最高级是双写该辅音字母然后再加 -er和-est。 big (原级) (比较级) (最高级) 6) 双音节和多音节形容词的比较级和最高级需用more 和 most 加在形容词前面来构成。 beautiful (原级) (比较级) (比较级) difficult (原级) (最高级) (最高级) 常用的不规则变化的形容词的比较级和最高级: 原级------比较级------最高级 good------better------best many------more------most much------more------most bad------worse------worst far------farther, further------farthest, furthest 形容词前如加 less 和 least 则表示"较不"和"最不 形容词比较级的用法: 形容词的比较级用于两个人或事物的比较,其结构形式如下: 主语+谓语(系动词)+ 形容词比较级+than+ 对比成分。也就是, 含有形容词比较级的主句+than+从句。注意从句常常省去意义上和主句相同的部分, 而只剩下对比的成分。

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

英语比较级和最高级的用法归纳

英语比较级和最高级的用法归纳 在学习英语过程中,会遇到很多的语法问题,比如比较级和最高级的用法,对于 这些语法你能够掌握吗?下面是小编整理的英语比较级和最高级的用法,欢迎阅读! 英语比较级和最高级的用法 一、形容词、副词的比较级和最高级的构成规则 1.一般单音节词和少数以-er,-ow结尾的双音节词,比较级在后面加-er,最高级 在后面加-est; (1)单音节词 如:small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest (2)双音节词 如:clever→cleverer→cleverest narrow→narrower→narrowest 2.以不发音e结尾的单音节词,比较在原级后加-r,最高级在原级后加-st; 如:large→larger→largest nice→nicer→nicest able→abler→ablest 3.在重读闭音节(即:辅音+元音+辅音)中,先双写末尾的辅音字母,比较级加-er,最高级加-est; 如:big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4.以“辅音字母+y”结尾的双音节词,把y改为i,比较级加-er,最高级加-est; 如:easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5.其他双音节词和多音节词,比较级在前面加more,最高级在前面加most; 如:bea utiful→more beautiful→most beautiful different→more different→most different easily→more easily→most easily 注意:(1)形容词最高级前通常必须用定冠词 the,副词最高级前可不用。 例句: The Sahara is the biggest desert in the world. (2) 形容词most前面没有the,不表示最高级的含义,只表示"非常"。 It is a most important problem. =It is a very important problem.

(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...

英语比较级和最高级的用法

More than的用法 A. “More than+名词”表示“不仅仅是” 1)Modern science is more than a large amount of information. 2)Jason is more than a lecturer; he is a writer, too. 3) We need more than material wealth to build our country.建设我们国家,不仅仅需要物质财富. B. “More than+数词”含“以上”或“不止”之意,如: 4)I have known David for more than 20 years. 5)Let's carry out the test with more than the sample copy. 6) More than one person has made this suggestion. 不止一人提过这个建议. C. “More than+形容词”等于“很”或“非常”的意思,如: 7)In doing scientific experiments, one must be more than careful with the instruments. 8)I assure you I am more than glad to help you. D. more than + (that)从句,其基本意义是“超过(=over)”,但可译成“简直不”“远非”.难以,完全不能(其后通常连用情态动词can) 9) That is more than I can understand . 那非我所能懂的. 10) That is more than I can tell. 那事我实在不明白。 11) The heat there was more than he could stand. 那儿的炎热程度是他所不能忍受的 此外,“more than”也在一些惯用语中出现,如: more...than 的用法 1. 比……多,比……更 He has more books than me. 他的书比我多。 He is more careful than the others. 他比其他人更仔细。 2. 与其……不如 He is more lucky than clever. 与其说他聪明,不如说他幸运。 He is more (a)scholar than (a)teacher. 与其说他是位教师,不如说他是位学者。 注:该句型主要用于同一个人或物在两个不同性质或特征等方面的比较,其中的比较级必须用加more 的形式,不能用加词尾-er 的形式。 No more than/not more than 1. no more than 的意思是“仅仅”“只有”“最多不超过”,强调少。如: --This test takes no more than thirty minutes. 这个测验只要30分钟。 --The pub was no more than half full. 该酒吧的上座率最多不超过五成。-For thirty years,he had done no more than he (had)needed to. 30年来,他只干了他需要干的工作。 2. not more than 为more than (多于)的否定式,其意为“不多于”“不超过”。如:Not more than 10 guests came to her birthday party. 来参加她的生日宴会的客人不超过十人。 比较: She has no more than three hats. 她只有3顶帽子。(太少了) She has not more than three hats. 她至多有3顶帽子。(也许不到3顶帽子) I have no more than five yuan in my pocket. 我口袋里的钱最多不过5元。(言其少) I have not more than five yuan in my pocket. 我口袋里的钱不多于5元。(也许不到5元) more than, less than 的用法 1. (指数量)不到,不足 It’s less than half an hour’s drive from here. 开车到那里不到半个钟头。 In less than an hour he finished the work. 没要上一个小时,他就完成了工作。 2. 比……(小)少 She eats less than she should. 她吃得比她应该吃的少。 Half the group felt they spent less than average. 半数人觉得他们的花费低于平均水平。 more…than,/no more than/not more than (1)Mr.Li is ________ a professor; he is also a famous scientist. (2)As I had ________ five dollars with me, I couldn’t afford the new jacket then. (3)He had to work at the age of ________ twelve. (4)There were ________ ten chairs in the room.However, the number of the children is twelve. (5)If you tel l your father what you’ve done, he’ll be ________ angry. (6)-What did you think of this novel? -I was disappointed to find it ________ interesting ________ that one. 倍数表达法 1. “倍数+形容词(或副词)的比较级+than+从句”表示“A比B大(长、高、宽等)多少倍” This rope is twice longer than that one.这根绳是那根绳的三倍(比那根绳长两倍)。The car runs twice faster than that truck.这辆小车的速度比那辆卡车快两倍(是那辆卡车的三倍)。 2. “倍数+as+形容词或副词的原级+as+从句”表示“A正好是B的多少倍”。

“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.

初中英语比较级和最高级讲解与练习

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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

英语比较级和最高级

形容词比较级和最高级的形式 一、形容词比较级和最高级的构成 形容词的比较级和最高级变化形式规则如下 构成法原级比较级最高级 ①一般单音节词末尾加 er 和 est strong stronger strongest ②单音节词如果以 e结尾,只加 r 和 st strange stranger strangest ③闭音节单音节词如末尾只有一个辅音字母, 须先双写这个辅音字母,再加 er和 est sad big hot sadder bigger hotter saddest biggest hottest ④少数以 y, er(或 ure), ow, ble结尾的双音节词, 末尾加 er和 est(以 y结尾的词,如 y前是辅音字母, 把y变成i,再加 er和 est,以 e结尾的词仍 只加 r和 st) angry Clever Narrow Noble angrier Cleverer narrower nobler angriest cleverest narrowest noblest ⑤其他双音节和多音节词都在前面加单词more和most different more different most different 1) The most high 〔A〕mountain in 〔B〕the world is Mount Everest,which is situated 〔C〕in Nepal and is twenty nine thousand one hundred and fourty one feet high 〔D〕 . 2) This house is spaciouser 〔A〕than that 〔B〕white 〔C〕one I bought in Rapid City,South Dakota 〔D〕last year. 3) Research in the social 〔A〕sciences often proves difficulter 〔B〕than similar 〔C〕work in the physical 〔D〕sciences. 二、形容词比较级或最高级的特殊形式:

高中英语的比较级和最高级用法总结

比较级和最高级 1.在形容词词尾加上―er‖ ―est‖ 构成比较级、最高级: bright(明亮的)—brighter—brightest broad(广阔的)—broader—broadest cheap(便宜的)—cheaper—cheapest clean(干净的)—cleaner—cleanest clever(聪明的)—cleverer—cleverest cold(寒冷的)—colder—coldest cool(凉的)—cooler—coolest dark(黑暗的)—darker—darkest dear(贵的)—dearer—dearest deep(深的)—deeper—deepest fast(迅速的)—faster—fastest few(少的)—fewer—fewest great(伟大的)—greater—greatest hard(困难的,硬的)—harder—hardest high(高的)—higher—highest kind(善良的)—kinder—kindest light(轻的)—lighter—lightest long(长的)—longer—longest loud(响亮的)—louder—loudest low(低的)—lower—lowest near(近的)—nearer—nearest new(新的)—newer—newest poor(穷的)—poorer—poorest quick(快的)—quicker—quickest quiet(安静的)—quieter—quietest rich(富裕的)—richer—richest short(短的)—shorter—shortest slow(慢的)—slower—slowest small(小的)—smaller—smallest smart(聪明的)—smarter—smartest soft(柔软的)—softer—softest strong(强壮的)—stronger—strongest sweet(甜的)—sweeter—sweetest tall(高的)-taller-tallest thick(厚的)—thicker—thickest warm(温暖的)—warmer—warmest weak(弱的)—weaker—weakest young(年轻的)—younger—youngest 2.双写最后一个字母,再加上―er‖ ―est‖构成比较级、最高级: big(大的)—bigger—biggest fat(胖的)—fatter—fattest hot(热的)—hotter—hottest red(红的)—redder—reddest sad(伤心的)—sadder—saddest thin(瘦的)—thinner—thinnest wet(湿的)—wetter—wettest mad(疯的)—madder—maddest 3.以不发音的字母e结尾的形容词,加上―r‖ ―st‖ 构成比较级、最高级:able(能干的)—abler—ablest brave(勇敢的)—braver—bravest close(接近的)—closer—closest fine(好的,完美的)—finer—finest large(巨大的)—larger—largest late(迟的)—later—latest nice(好的)—nicer—nicest ripe(成熟的)—riper—ripest

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.

(完整版)初中英语比较级和最高级的用法

英语语法---比较级和最高级的用法 在英语中通常用下列方式表示的词:在形容词或副词前加more(如 more natural,more clearly )或加后缀 -er(newer,sooner )。典型的是指形容词或副词所表示的质、量或关系的增加。英语句子中,将比较两个主体的方法叫做“比较句型”。其中,像“A比B更……”的表达方式称为比较级;而“A最……”的表达方式则称为最高级。组成句子的方式是将形容词或副词变化成比较级或最高级的形态。 一、形容词、副词的比较级和最高级的构成规则 1.一般单音节词和少数以-er,-ow结尾的双音节词,比较级在后面加-er,最高级在后面加-est; (1)单音节词 如:small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest (2)双音节词 如:clever→cleverer→cleverest narrow→narrower→narrowest 2.以不发音e结尾的单音节词,比较在原级后加-r,最高级在原级后加-st; 如:large→larger→largest nice→nicer→nicest able→abler→ablest 3.在重读闭音节(即:辅音+元音+辅音)中,先双写末尾的辅音字母,比较级加-er,最高级加-est; 如:big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4.以“辅音字母+y”结尾的双音节词,把y改为i,比较级加-er,最高级加-est; 如:easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5.其他双音节词和多音节词,比较级在前面加more,最高级在前面加most; 如:beautiful→more beautiful→most beautiful different→more different→most different easily→more easily→most easily

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