a r
X
i
v
:07
5
.
5
2
v
1
[
q
u
a n
t
-
p
h
]
3
M
a y
2
7
Slow cross-symmetry phase relaxation in complex collisions Luis Benet,1Lewis T.Chadderton,2Sergey Yu.Kun,3,4,?Oleg K.Vorov,5and Wang Qi 61Instituto de Ciencias F′?sicas,Universidad Nacional Aut′o noma de M′e xico (UNAM),62210–Cuernavaca (Morelos),Mexico 2Atomic and Molecular Physics Laboratary,RSPhysSE,Australian National University,Canberra ACT 0200,Australia 3Facultad de Ciencias,Universidad Aut′o noma del Estado de Morelos,62209–Cuernavaca (Morelos),M′e xico 4Nonlinear Physics Center and Department of Theoretical Physics,RSPhysSE,Australian National University,Canberra ACT 0200,Australia 5Department of Physics and Astronomy,Drake University,Des Moines,Iowa 50311,USA 6Institute of Modern Physics,Chinese Academy of Sciences,Lanzhou 730000,China We discuss the e?ect of slow phase relaxation and the spin o?-diagonal S -matrix correlations on the cross section energy oscillations and the time evolution of the highly excited intermediate systems formed in complex collisions.Such deformed intermediate complexes with strongly overlapping resonances can be formed in heavy ion collisions,bimolecular chemical reactions and atomic cluster collisions.The e?ects of quasiperiodic energy dependence of the cross sections,coherent rotation of the hyperdeformed ?(3:1)intermediate complex,Schr¨o dinger cat states and quantum-classical transition are studied for 24Mg+28Si heavy ion scattering.
PACS numbers:25.70.-z,https://www.doczj.com/doc/3514040064.html,,03.65.Nk,03.65.-w
I.INTRODUCTION
The dominating idea in the modern theory of highly excited strongly interacting systems is that phase randomization time is the shortest time scale of the problem [1].Applica-
tion of this idea to the theory of quantum chaotic scattering for colliding systems with rotationally invariant Hamiltonians implies the absence of correlations between the reaction amplitudes carrying di?erent total spins[1].However,though seemingly plausible,the as-sumption of a very fast phase relaxation is not consistent with many data sets on complex quantum collisions.In particular,the anomalously long-lived spin o?-diagonal S-matrix correlations have been identi?ed from the data on forward peaking of evaporating protons in nucleon induced[2,3]and photonuclear[4]reactions.Such long-lived correlations re?ect an anomalously slow phase relaxation,which is many orders of magnitude longer than the energy relaxation.This provides a manifestation of a new form of matter:thermalized non-equilibrated matter introduced by one of us in Refs.[5,6].The e?ect is of primary importance for many-qubit quantum computation since anomalously long“phase memory”can extend the time for quantum computing far beyond the quantum chaos border[2,4]. An e?ect of a very slow phase relaxation has also been strongly supported by numerical calculations for H+D2[7],F+HD[8]and He+H+2[9]state-to-state chemical reactions.In these calculations,a slow phase relaxation manifests itself in stable rotating wave packets of the intermediate complexes[10].Interestingly,this same e?ect of stable coherent rota-tion was originally revealed for heavy ion collisions,e.g.,for19F+89Y[11],28Si+64Ni[12], 12C+24Mg[13,14],24Mg+24Mg and28Si+28Si[15],58Ni+46Ti and58Ni+62Ni[16]collisions.
In this paper we reveal the e?ect of slow phase relaxation for yet another heavy ion scattering system,24Mg+28Si.This e?ect is studied in relation to quasiperiodic energy de-pendence of the cross sections,coherent rotation of the hyperdeformed?(3:1)intermediate complex,Schr¨o dinger cat states and quantum-classical transition for24Mg+28Si heavy ion scattering.
II.CROSS SECTION ENERGY AUTOCORRELATION FUNCTION
We consider?rst the scattering of spinless collision partners with spinless collision fragments in the exit https://www.doczj.com/doc/3514040064.html,ing the semiclassical asymptotics of Legendre polynomials for J?1, where J is the total spin of the system,we represent the cross section in the form
dσ(E,θ)/dθ≡σ(E,θ)=σd(θ)+δσ(E,θ),(1)
withδσ(E,θ)=δσ(+)(E,θ)+δσ(?)(E,θ),δσ(±)(E,θ)=|δf(±)(E,θ)|2and
δf(±)(E,θ)= J(2J+1)W(J)1/2δˉS J(E)exp[iJ(Φ±θ)].(2) In these expressions,σd(θ)=|F d(θ)|2is the energy independent potential scattering cross section,Φis the average de?ection angle obtained from a linear approximation for the J-dependence of the potential phase shifts in the entrance and exit channels,W(J)is the average partial reaction probability,andδˉS J(E)are normalized, |δˉS J(E)|2 =1,energy ?uctuating around zero S-matrix elements corresponding to time-delayed collision processes. The brackets ... stand for the energy E averaging.In the expression forσ(E,θ)we have dropped(i)the highly oscillating angle interference term between theδf(+)(E,θ)and δf(?)(E,θ)amplitudes,and(ii)the interference terms between the energy smooth potential scattering amplitude F d(θ)and energy?uctuating amplitudesδf(±)(E,θ).This is because the excitation functions data for24Mg+28Si scattering[17]were obtained by averaging over a wide?θc.m.?77??98?angular range.
In calculating the cross section energy autocorrelation function,
C(ε)= σ(E+ε,θ)σ(E,θ) / σ(E,θ) 2?1,(3) we take into account the S-matrix spin o?-diagonal correlation[18]
δˉS J(E+ε)δˉS J′(E)? =Γ/(Γ+β|J?J′|+iˉhω(J?J′)?iε).(4)
Here,ωis the angular velocity of the coherent rotation of the intermediate complex,βis the spin phase relaxation width andΓis the total decay width of the intermediate complex.
We take W(J)in the J-window form,W(J)=W(|J?I(E)|/g),where the average spin I(E),for a given c.m.energy of the collision partners,is close to the grazing orbital momentum.The J-window width g relates to the e?ective number of partial waves,g+1, contributing toδσ(E,θ).For the analyzed24Mg+28Si scattering we estimate g?1?5, which is revealed by the shape of the measured elastic scattering angular distributions[17] corresponding to the maxima of the excitation functions.Although these angular distri-butions show regular oscillations with a well-de?ned period,they clearly deviate from the square of a single Legendre polynomial.We estimate the energy dependence of I(E)in the linear approximation and obtain I(E)=ˉI+ˉI(E?ˉE)/?E.Here,ˉI=I(ˉE),ˉE is the energy corresponding to the center of the energy interval over which the cross section is measured,
?E=2(E?B)/ˉI and B is the Coulomb barrier for the collision partners in the entrance channel for a con?guration of the two touching spherical nuclei24Mg and28Si.
We calculate C(ε)under the conditions g≥1,β≤Γandβ?ˉhω.We take W(J)in the Gaussian form,W(J)∝exp[?(J?I(E))2/g2].Generalizing the calculations in[19]to the case of?niteβand arbitrary?E for the normalized(C(ε=0)=1)cross-section energy autocorrelation function,we obtain
C(ε)=exp[?ε2/2(ˉhω)2d2]
1?exp[iπ(|ε|+iΓ)/(ˉhω?iβ)]
,(5)
where d2=g2/(1?ˉhω/?E)2.The above expression for C(ε)has been obtained for d≥1. One can see thatβ/ˉh has the physical meaning of the imaginary part of the angular velocity signifying the space-time delocalization of the nuclear molecule and the damping of the coherent rotation of the intermediate complex[18].Forβ=0,C(ε)is an oscillating periodic function with periodˉhω.For?niteβ,the amplitude of the oscillations in C(ε)decreases with increasing|ε|:the largerβthe stronger the damping of the oscillations.Forβ=0and ˉhω=?E,Eq.(5)transforms to the result in Ref.[19].
Although Eq.(5)is obtained for spinless reaction fragments it also holds for reaction products having intrinsic spins.This can be shown using the helicity representation for the scattering amplitude[18].
In Fig.1we present the energy autocorrelation functions for the24Mg+28Si elastic and inelastic scattering[20],constructed from the data on the excitation functions measured on the E c.m.=49?57MeV energy interval[17].The experimental excitation functions were averaged on theθcm=77??98?angle interval.The experimental C(ε)’s are not Lorentzian but oscillate with a period?0.75MeV.The?t of the experimental C(ε)’s for all the elastic and inelastic channels is obtained withΓ=0.15MeV,β=0.1MeV,ˉhω=0.75MeV and d=5.The calculated C(ε)’s are normalized to the experimental data atε=0.
The extracted value ofˉhωsuggests an anomalously strong deformation of the interme-diate complex.Indeed,for J?34?38[17],using the moment of inertia of a?(2:1) superdeformed intermediate complex,corresponding to the two touched spherical colliding nuclei24Mg and28Si,we haveˉhω?1.9MeV.This value is bigger by a factor of about2.5 than the period of oscillations in the experimental C(ε)’s.This reveals the excitation of ?(3:1)hyperdeformed coherent rotational states of the intermediate complex.
It should be noted that the intrinsic excitation energy of the intermediate complex,which
we obtain by substracting deformation and rotation energy for the total energy,is about 15MeV or more.This corresponds to the average level spacing of D?10?6MeV or less. Therefore,the intermediate complex is in the regime of strongly overlapping resonances,Γ/D≥105.In this regime,the theory of quantum chaotic scattering and random matrix theory are conventionally assumed to apply[1].In accordance with these approaches,which in particular recon?rmed the Ericson theory of the compound nucleus cross-section?uctua-tions[21],the spin o?-diagonal S-matrix correlations vanish yielding C(ε)=1/[1+(ε/Γ)2]. The Lorenzian curves presented in Fig.1withΓ=0.85MeV to?t the experimental data at≤0.1MeV are clearly in contrast with the oscillations in all the experimental C(ε)’s. Within our approach,the limit of vanishing spin o?-diagonal S-matrix correlations corre-sponds toβ?Γ,whereˉh/βis the characteristic spin phase relaxation time.Therefore, the persistence of the oscillations in C(ε)indicates an anomalously long spin o?-diagonal “phase memory”.
In Fig.1we present another possible?t of the experimental C(ε)’s with the sameΓ=0.15 MeV andˉhω=0.75MeV,but with the di?erent valuesβ=0.03MeV and d=1.One can see that both the?ts are qualitatively and quantitatively undistinguishable.The question arises if the quantities of the interest,in particular the phase relaxation widthβ,can reliably be determined from the data.
III.TIME POWER SPECTRUM OF THE COLLISION
Consider the time(t)power spectrum of the collision for the spinless reaction partners in the entrance and exit channels.Unlike the cross section energy autocorrelation function in the previous Section,the time power spectrum will be studied for a?ne angular resolution. The time power spectrum is given by the Fourier component of the amplitude energy au-tocorrelation function[14,21].For t?ˉh/D,i.e.forΓ/D?1,the continuous spectrum approximation is valid and we have[10,18,22]
P(t,θ)∝H(t)exp(?Γt/ˉh) JJ′[W(J)W(J′)]1/2exp[i(Φ?ωt)(J?J′)?β|J?J′|t/ˉh]P J(θ)P J′(θ).
(6) Here,P J(θ)are Legendre polynomials,and the Heaviside step function H(t)signi?es that the intermediate complex cannot decay before it is formed at t=0.
In Fig.2we present P(t,θ)for three moments of time and for the two di?erent sets of
the parameters for which the C(ε)’s were calculated in the previous Section(Fig.1).The ?rst set is:Γ=0.15MeV,ˉhω=0.75MeV,ˉI=36,Φ=0,β=0.03MeV,d=1.For the second set we have di?erent values ofβ=0.1MeV,d=5while the rest of the parameters is unchanged.For the reason discussed in[10]the time power spectra in Fig.2are scaled with the P diag(t,θ),which is given by Eq.(6),where only the spin diagonal terms J=J′are taken into account.Such a spin diagonal approximation corresponds to the limit of quantum chaotic scattering and random matrix theory[1].Accordingly,deviation of the scaled time power spectra in Fig.2from a constant unity is a quantitative measure of the deviation of the collision process from the universal limit of the quantum chaotic scattering theory[1].
Fig.2illustrates a rotation of the two wave packets towards each other.As the wave packets rotate they also spread-the biggerβthe faster the spreading.One observes that, forβ=0.03MeV and d=1,the contrast of the interference fringes,due to the interfer-ence between the near-side and far-side amplitudes[10],is very strong.These interference fringes is a manifestation of Schr¨o dinger cat states in highly excited quantum many-body systems[22].On the contrary,forβ=0.1MeV and d=5,the contrast of the inter-ference fringes is greatly reduced indicating a quantum-classical transition in the collision process[10].
Our approach shows that the complicated many-body collision problem can be accurately represented by the simple picture of a weakly damped(β<<ˉhω)quantum rotator.This picture was obtained without introducing any collective degrees of freedom of the interme-diate complex,such as its deformation and spatial orientation.The introduction of those degrees of freedom is known to be a successful approximation[23]for very low,closed to Yrast line,intrinsic excitations of the intermediate complex.Yet,in our case of high in-trinsic excitations(≥15MeV),the collective degrees of freedom acquire large spreading widths[24],Γspr>>β,Γ,and by consequence they decay much faster than the average life-time of the intermediate complex.
Notice that P(t,θ)can be obtained from the data for excitation function?uctuations for binary collisions,for?ne energy and angular resolutions,provided the relative contribution of direct processes is signi?cant(≥70%)[25].The latter is usually the case for heavy-ion elastic and inelastic scattering.Experimentally,?ne energy[25]and angular[26]resolutions required for the determination of P(t,θ)are routinely achievable for heavy ion collisions[27]. Therefore,a reliable determination of the phase relaxation widthβ,which is an important
new energy scale in quantum many-body systems[2,4,10],is experimentally possible.
IV.CONCLUSION
We have discussed the e?ects of slow phase relaxation,spin o?-diagonal S-matrix correla-tions on the cross section energy oscillations and the time evolution of the highly excited intermediate system formed in the24Mg+28Si collision.Such quasiperiodic energy oscilla-tions were observed experimentally.The e?ects of coherent rotation of the hyperdeformed ?(3:1)intermediate complex,Schr¨o dinger cat states and quantum-classical transition have been revealed for the24Mg+28Si heavy ion scattering.
L.B.and S.Yu.K.acknowledge with gratitude?nancial support from the projects IN-101603(DGAPA-UNAM)and43375(CONACyT).
[1]T.Guhr,A.M¨u ller-Groeling,and H.A.Weidenm¨u ller,Phys.Rep.299,189(1998),and ref-
erences therein.
[2]J.Flores,S.Yu.Kun,and T.H.Seligman,Phys.Rev.E72,017201(2005);quant-ph/0502050.
[3]M.Bienert,J.Flores,and S.Yu.Kun,Phys.Rev.C74,027602(2006);nucl-ex/0508020.
[4]M.Bienert,J.Flores,S.Yu.Kun,and T.H.Seligman,SIGMA2,Paper027(2006);
quant-ph/0602224.
[5]S.Yu.Kun,Z.Phys.A348,273(1994).
[6]S.Yu.Kun,Z.Phys.A357,255(1997).
[7]S.C.Althorpe,F.Fern′a ndez-Alonso,B.D.Bean,J.D.Ayers,A.E.Pomerantz,R.N.Zare,and
E.Wrede,Nature(London)416,67(2002).
[8]S.C.Althorpe,J.Phys.Chem.A107,7152(2003).
[9] A.N.Panda and S.C.Althorpe,Chem.Phys.Lett.419,245(2006).
[10]L.Benet,L.T.Chadderton,S.Yu.Kun,and Wang Qi,“Quantum-classical transition for an
analog of double-slit experiment in complex collisions:Dynamical decoherence in quantum many-body systems”,quant-ph/0610091.
[11]S.Yu.Kun et al.,Z.Phys.A359,263(1997).
[12]S.Yu.Kun,V.Yu.Denisov,and A.V.Vagov,Z.Phys.A359,257(1997).
[13]S.Yu.Kun,A.V.Vagov,and O.K.Vorov,Phys.Rev.C59,R585(1999).
[14]S.Yu.Kun,A.V.Vagov,and W.Greiner,Phys.Rev.C63,014608(2001).
[15]S.Yu.Kun,B.A.Robson,and A.V.Vagov,Phys.Rev.Lett.83,504(1999).
[16]S.Yu.Kun et al.,Z.Phys.A359,145(1997).
[17] A.H.Wuosmaa,S.Saini,P.H.Kutt,S.F.Pate,R.W.Zurm¨u hle,and R.R.Betts,Phys.Rev.
C36,1011(1987).
[18]S.Yu.Kun,Z.Phys.A357,271(1997).
[19]S.Yu.Kun,Europhys.Lett.26,505(1994).
[20] A.Sarma and R.Singh,Z.Phys.A337,23(1990).
[21]T.Ericson and T.Mayer-Kuckuk,Ann.Rev.Nucl.Sci.16,183(1966).
[22]S.Yu.Kun,L.Benet,L.T.Chadderton,W.Greiner,and F.Haas,Phys.Rev.C67,011604(R)
(2003);quant-ph/0205036.
[23] A.Bohr and B.R.Mottelson,Nuclear Structure(Benjamin,New York,1969).
[24]O.Bohigas and H.A.Weidenm¨u ller,Annu.Rev.Nucl.Part.Sci.38,421(1988).
[25]L.Benet,S.Yu.Kun,and Wang Qi,Phys.Rev.C73,064602(2006);quant-ph/0503046.
[26]L.Benet,S.Yu.Kun,Wang Qi,and V.Denisov,Phys.Lett.B605,101(2005);
nucl-th/0407029.
[27]R.R.Betts and A.H.Wuosmaa,Rep.Prog.Phys.60,819(1997).
FIG.1:Experimental(dots)and calculated C(ε)’s for24Mg+28Si elastic and inelastic scattering. Dashed lines are obtained with d=1andβ=0.03MeV,and dashed-dotted lines with d=5and β=0.1MeV(see text).Dotted lines are Lorentzians withΓ=0.085MeV.
FIG.2:The time power spectra for the24Mg+28Si scattering obtained for the three moments of time with T being a period of one complete revolution of the intermediate complex.Solid thin lines are obtained with d=1andβ=0.03MeV,and solid thick lines with d=5andβ=0.1MeV
(see text).
比较级和最高级 1.用“as+原级+as”表示 Tom is as tall as Mike. 2.用“not as(so) +原级+as”或“less than”表示 I didn’t do my homework so(as) carefully as you. The picture is less attractive than that one. 3.用“比较级+than”表示 Our city is more beautiful than any other city in our country. 注意：1) 为了避免重复，在从句中常用one, that, those等词来代替前面提过的名词。 The weather here is warmer than that of Shanghai. The radios made in our factory are better than those in your factory. 2)比较等级应注意避免和包括自己的对象比。 比较级+than+ any other + 单数名词 all the other + 复数名词 anyone else any of the other + 复数名词 3）如果形容词作定语修饰一个单数可数名词，一般将不定冠词a/an放在形容词之后。 Our neighbour has _____ ours. A. as a big house as
B. as big a house as C. the same big house as D. house the same big as 4）比较级前一般不用冠词，但若表示“两者中较……时”。比较级前要加定冠词。若比较级后有名词，常在比较级前加不定冠词，表示泛指。 E.g. 他是两者中较高的一个 He is the taller of the two. 她唱得真动听！我可从未听过比这更好的嗓音了。 How beautifully she sings! I have never heard a better voice. 4. 三者或三者以上相比，表示最高级时，用“the +最高级”的结构表示，这种句式一般常有表示比较范围的介词短语。 Zhang Hua is the tallest of the three. He works (the) hardest in his class. That was the least exciting football game I’ve ever watched. This hotel is the most comfortable I’ve ever stayed. 注意：当最高级的前面无限定词the或有不定冠词a/an时，仅表示“很……，非常……” Monday is my busiest day. 星期一是我很忙的一天。 Qingdao is a most (very) beautiful coastal city. 青岛是一个非常美丽的海滨城市。 一、请写出下列形容词的比较级和最高级。 big ______ ______ small ______ ________ new ____?__ ________ tall ______ ______ short______ ________ old____?__ ________ weak ______ ______ strong ______ ______ fat____?__ ________ hot ______ ______ cold ______ ________ thin ____?__ ________ nice ______ _____ good ______ ________ high____?__ ________ low____?__ ________cheap______ ______ easy ______ ________
形容词、副词的比较级和最高级的用法： 当两种物体之间相互比较时，我们要用形容词或副词的比较级； 当相互比较的物体是三个或三个以上时，我们就要用形容词或副词的最高级。※形容词、畐I」词的比较级和最高级的变化规律： 1. 单音节形容词或副词后面直接加-er或-est tall —taller —tallest fast —faster —fastest 2. 以-e结尾的单音节形容词或副词直接加-r或-st large —larger —largest n ice —ni cer —ni cest 3. 以-y结尾的形容词或副词，改-y为-i再加-er或-est busy—busier —busiest early —earlier —earliest 4. 形容词或副词是重读闭音节时，双写最后的辅音字母，再加-er或-est hot ——hotter — hottest big ——bigger — biggest 5. 多音节形容词或副词前面直接加more或most delicious —more delicious —most delicious beautiful ——more beautiful ——most beautiful 6. 不规则变化 good (well) —better —best bad (badly) —worse—worst man y(much)-more-most little-less-least old-older(elder)-oldest(eldest) far-farther(further)-farthest(furthest) 以下笔记请手动记录！！！
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→narro west 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 e asily 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. 6.有少数形容词、副词的比较级和最高级是不规则的，必须熟记。 如：good→better→best well→better→best bad→worse→worst ill→worse→worst
比较级和最高级的讲解 变化规则 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 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"。
一、比较级和最高级的讲解 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→narrowes t 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 e asily 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. 6．有少数形容词、副词的比较级和最高级是不规则的，必须熟记。 如：good→better→best well→better→best bad→worse→worst ill→worse→worst old→older/elder→oldest/eldest many/much→more→most little→less→least far →further/farther→ furthest/farthest 二、形容词、副词的比较级和最高级的用法 1．“A + be +形容词比较级+ than + B” 意思为“A比B更……”。 如：This tree is taller than that one. 这棵树比那棵树高。 注意： ①在含有连词than的比较级中，前后的比较对象必须是同一范畴，即同类事物之间的比较。 ②在比较级前面使用much，表示程度程度“强得多”。 如：A watermelon is much bigger than an apple. ③very, quite一般只能修饰原级，不能修饰比较级。 2．“比较级+ and + 比较级”或“more and more +原级”表示“越来越……”
形容词比较级和最高级 绝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 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 old ----older,elder----older,eldest 形容词前如加 less 和 least 则表示"较不"和"最不" important 重要 less important 较不重要 least important 最不重要 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。
形容词比较级、最高级变化表 一、形容词比较级、最高级变化规则 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 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 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 4.以字母y结尾的形容词，把y改为i，再加上“er” “est” 构成比较级、最高级：busy（忙碌的）—busier—busiest dirty（脏的）—dirtier—dirtiest dry（干燥的）—drier—driest early（早的）—earlier—earliest 5.双音节、多音节形容词，在单词前面加上“more” “most” 构成比较级、最高级：afraid（害怕的）—more afraid—most afraid beautiful（美丽的）—more beautiful—most beautiful 6.不规则变化的形容词： bad（坏的）—worse—worst far（远的）—farther—farthest (far—further—furthest) good（好的）—better—best ill（病的）—worse—worst
形 容 词 的 比 较 级 和 最 高 级 变 化 规 则 B.部分双音节与多音节的词比较级在原级之前加more, 最高级在原级之前 加most beautiful---more beautiful---most beautiful interesting--- difficult--- C.不规则变化的形容词: little / few（原形）- less （比较级）- least（最高级） good（原形）- better（比较级）- best（最高级） bad （原形）- worse（比较级）- worst（最高级） far （原形）-- further—furthest 例句： Tom is tall. John is tall. Bob is tall. I'm as tall as you. Tom is as tall as John.
Bob is taller than John. John is the tallest of the three. John is the tallest in his class. 写出以下各形容词的比较级和最高级： 1. nice ______________________ 2. fat ____________________ 3. slow _____________________ 4. dry ____________________ 5. happy ____________________ 6. wet ____________________ 7. much ____________________ 8. ill _____________________ 9. little _____________________ 10. bad ___________________ 11. thin ______________________ 12. far ____________________ 13. early _____________________ 14. careful_________________ 15. exciting ___________________ 16. busy __________________ 2. 根据句意，用所括号内所级形容词的比较等级形式填空： 1. Mr. Smith is _________ man in this office. (rich) 2. Winter is _________ season of the years. (cold) 4. It is much _______ today than yesterday. (hot) 5. She is a little ________ than her classmates. (careful) 6. ________ people came to the meeting than last time. (many) 7. Which book is ________, this one or that one? (easy) 8. My room is _______ than yours. (small) 9. Hainan is _______ from Beijing than Hunan. (far) 10. Skating is _______ than swimming. (exciting) 11. Jim is _______ than all the others. (honest) 12. Things are getting _______ and _______. (bad) 13. The higher you climb, the _______ it will be. (cold) 14. Now his life is becoming ________ and _______. (difficult) 用适当形式填空： 1. Bob is _________ ( young ) than Fred. but ___________ (tall) than Fred. 2. Almost all the students' faces are the same ,but Li Deming looks _______ (fat) than before after the summer holidays. 3.Which is _________ (heavy), a duck or a chicken? 5.-- How _________ (tall) is Sally? --She' s 1.55 metres ________ (tall). What about Xiaoling? -- She' s only 1.40 metres ________ (tall). She is much _______ (short) than Sally. She is also the _______ (short) girl in the class. 6. He is ______ (bad) at learning maths. He is much _______ (bad) at Chinese and he is the _________ (bad) at English. 7. Annie says Sally is the ________ (kind) person in the world. 8. He is one of the_________(friendly) people in the class, I think. 9. A dictionary is much _________ (expensive) than a story-book. 10. An orange ia a little ______ (big) than an apple, but much ________ (small) than a watermelon.
形容词 第一章比较级、最高级变化一览表 规则变化 1.单音节以及少数双音节的词尾加上“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以一个元音加一个辅音字母结尾的单音节词（即重读闭音节词），双写结尾的辅音字母er， -est 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 特别提醒：new, few, slow, clean等词含有字母组合，且发的是长元音，不用双写。 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 rude（粗鲁的）—ruder—rudest safe（安全的）—safer—safest strange（奇怪的）—stranger—strangest wide（宽广的）—wider—widest wise（睿智的，聪明的）—wiser—wisest white（白的）—whiter—whitest 4.“以辅音字母+y”结尾的词改y为i，再加-er, -est busy（忙碌的）—busier—busiest dirty（脏的）—dirtier—dirtiest
比较级和最高级 一、比较级的用法: 当两个人或事物（A和B）进行比较时，我们需要用到形容词（副词）的原级或者比较级1．表达“A和B一样”，用as…as的结构。 公式: A+be动词+as+形容词原级+as…+B A+实义动词+as+副词原级+as…+B Eg I am as tall as you. He runs as fast as I. 我的房间和她的一样大。 他游得和我一样好。 2．表达“A不如B”用not as…as的结构。 公式: A+be动词的否定形式+as+形容词原级+as…+B A+助词的否定形式+动词+as+形容词原级+as…+B Eg I am not as tall as you. He doesn’t run as fast as I. 我的房间没有他的大。 我没有他游得好。 3. 表达“A大于B”用“比较级+than”的结构。 公式: A+be动词+形容词比较级+than+B… A+实义动词+副词比较级+than+B… Eg I am taller than you.我比你高。 He runs faster than I. 他跑得比我快。 我的房间比他的大。 我游得比他的好。 4.表示A 是...中最大的结构 公式：A+be动词+the +形容词最高级+范围 A+实义动词+the+形容词最高级+范围 I am the tallest in my class. He runs the fastest in my class. 我的房间是这里最大的。 我游得是我们班最好的。 二.形容词和副词的比较级和最高级的变化方法如下
(1) 符合规则的： (2)几个不规则的形容词和副词的比较级和最高级如下表： 原 级 比较级 最高级 good , well better best bad , ill worse worst many , much more most little less least far farther / further farthest / furthest 练习1：写出下列词的比较级和最高级 tall ﹍﹍ ﹍﹍ slow ﹍﹍ ﹍﹍ small ﹍﹍ ﹍﹍ fast ﹍﹍ ﹍﹍ smart ﹍﹍ ﹍﹍ few ﹍﹍ ﹍﹍ nice ﹍﹍ ﹍﹍ fine ﹍﹍ ﹍﹍ large ﹍﹍ ﹍﹍ late ﹍﹍ ﹍﹍ brave ﹍﹍ ﹍﹍ pretty ﹍﹍ ﹍﹍ easy ﹍﹍ ﹍﹍ funny ﹍﹍ ﹍﹍ happy ﹍﹍ ﹍﹍ lazy ﹍﹍ ﹍﹍ heavy ﹍﹍ ﹍﹍ dirty ﹍﹍ ﹍﹍ dry ﹍﹍ ﹍﹍ early ﹍﹍ ﹍﹍ 情 况 加 法 例 词 一 般 情 况 直接加 -er ; -est all-taller-tallest 以不发音e 结尾的词 去e 加 –er ; -est nice-nicer-nicest 以“辅音+y”结尾的词 变y 为i 再加-er ; -est dry-drier-driest heavy-heavier-heaviest 重读闭音节结尾的词 双写末尾辅音字母，再加-er ; -est thin-thinner-thinnest 多音节和部分双音节单词 在词前加 more ; most more delicious most delicious
. 比较级和最高级列表 good-better-best new-newer-newest bad/ill-worse-worst far-farther-farthest far-further-furthest many/much-more-most little-less-least long-longer-longest young-younger-youngest old-older/elder-oldest/eldest short-shorter-shortest high-higher-highest deep-deeper-deepest small-smaller-smallest big-bigger-biggest tall-taller-tallest loud-louder-loudest low-lower-lowest thin-thiner-thinest fat-fatter-fattest great-greater-greatest nice-nicer-nicest happy-happier-happiest heavy-heavier-heaviest cheap-cheaper-cheapest near-nearer-nearest clean-dleaner-cleanest few-fewer-fewest late-later-latest angry-angrier-angriest busy-busier-busiest lazy-lazier-laziest hot-hotter-hottest glad-gladder-gladdest clear-clearer-clearest strong-stronger-strongest lucky-luckier-luckiest interesting-more interesting -most interesting difficult-more difficult-most difficult expensive-more expensive -most expensive
形容词的比较级和最高级 (2007-04-10 10:03:15) 转载 分类：教育教学 形容词的比较级和最高级 英语中大多数形容词是可以分级的，一般有三个等级：原级，比较级和最高级。 原级 形容词的本来形式就是形容词的原级。用原级进行比较时可以使用下面两种结构： 1. 表示比较的双方相等，用”as…as”结构，表示“前者像后者一样”，即 A + be + as + 形容词原级 + as + B. e.g. John is as tall as his brother. 2. 表示比较的双方不相等，用”not as…as”结构，表示“前者不如后者”，即 A + be + not as + 形容词原级 + as + B. e.g. John is not as tall as his brother. 比较级 当把一个人或物同另外一个人或物比较时，就需要用到形容词比较级。其结构是： A + be + 形容词比较级 + than + B. e.g. Jackei is taller than Alex, but Alex is heavier than Jackei
最高级 三者或者三者以上的人或物进行比较时，需要使用形容词的最高级。其结构是： A + be + the 形容词最高级 + of / in + 比较范围. (Note: 形容词最高级前面要加定冠词 the) e.g. Jackei is the tallest in our class. or Jackei is the tallest of all the students. 比较级的几种用法： 1，表示倍数， A+be+数词+times+形容词比较级+than+B Our room is twice larger than theirs. 我们的房间是他们的两倍大 注：两倍为twice而不是two times. 2，表示大多少，多多少，高多少等 A+be+数量词+比较级+than+B She is two years older than me 她比我大两岁 3，用比较级表示最高级 A+be+比较级+than+any other+单数名词，或者，A+be+比较级+than+the other+复数名词
小学英语常见形容词及比较级、最高级变化一览表 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 young（年轻的）—younger—youngest 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 deep（深的）—deeper—deepest warm（温暖的）—warmer—warmest 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 strong（强壮的）—stronger—strongest weak（弱的）—weaker—weakest sweet（甜的）—sweeter—sweetest tall（高的）-taller-tallest thick（厚的）—thicker—thickest 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 wet（湿的）—wetter—wettest thin（瘦的）—thinner—thinnest 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： 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 safe（安全的）—safer—safest strange（奇怪的）—stranger—strangest wide（宽广的）—wider—widest 4.以字母y结尾的形容词，把y改为i，再加上“er”“est”构成比较级、最高级：busy（忙碌的）—busier—busiest dirty（脏的）—dirtier—dirtiest dry（干燥的）—drier—driest early（早的）—earlier—earliest easy（容易的）—easier—easiest friendly（友好的）—friendlier—friendliest funny（好玩的）—funnier—funniest happy（开心的）—happier—happiest healthy（健康的）—healthier—healthiest heavy（重的）—heavier—heaviest
形容词的比较级和最 高级教案
形容词的比较级和最高级 Teach ing objectives: 1. Knowledge goals ①规则形容词原级变化比较级与最高级的方法 ②用含有比较级与最高级的句子来描述事物 2. Ability goals ①基本掌握规则形容词原级变化比较级与最高级的方法 ②用含有比较级与最高级的句子来描述图片和发表观点 Teach ing focus： 1. The comparative and superlative of adjectives. 2. How to use comparative and superlative degrees to compare things. Teach ing difficulties: 1. 基本掌握句型“A is…than B” 2基本掌握句型“A is the…of all. ” Teach ing methods: 直观教学法、任务教学法和归纳法。
Teach ing aids: Multi-media Teach ing procedures: Stepl. Pre-task 1. Lead in g-i n Use food to lead in the comparative and superlative degrees. 2. Guide Ss to know how to change adjectives into comparative and superlative degrees. 3. Practice Step2. While-task 1. Show some pictures and ask Ss to describe them using “A is … than B” . 2. Practice 3. Ask Ss to use “ A is the …of all.” to describe the pictures. 4. Pair work Step3. Post-task Give a situati on,l et the Ss choose the best way to travel to sp. Tips: fast—slow cheap— expe nsive S1: Which is the ____ o f all?/ I want to go there by ____ . S2: The ____ is the ____ of all./ Because the ____ i s ____ than the ____ Step4. Homework Write 8 senten ces using the comparative and superlative degrees.